Are there any scholars or doctrinal schools that address the conspicuous dearth of needy people in Jesus' sayings and stories?

I'm a bit embarrassed that this thought did not occur to me sooner, given my moniker on this website (viz., "rhetorician"). That thought is simply this: If I'm the Messiah (stay with me here!), and a big part of my commission by God is to

"PREACH THE GOSPEL TO THE POOR. . . . TO PROCLAIM RELEASE TO THE CAPTIVES, AND RECOVERY OF SIGHT TO THE BLIND, TO SET FREE THOSE WHO ARE OPPRESSED, TO PROCLAIM THE FAVORABLE YEAR OF THE LORD" (Luke 4:18-19, excerpts NASB Updated),

then as God's spokesman who is anointed by him to preach his good news to the poor, I would then preach to the poor.

By this I do not mean I would turn my back on the rich or discourage them from embracing my message. No, it means that my primary target audience (again, let's think rhetorically, as a public speaker/preacher/teacher should) is primarily to the poor, yet not to the exclusion of the rich.

Why, then, would Jesus use more "rich" characters in his parables than "poor characters"? Simply because of the difficulty rich people have, generally speaking, to trust God for their daily needs (e.g., food, water, clothes, shelter) and then be content with having just those needs met and no more. By using more rich people in his parables he is emphasizing or drawing attention to, perhaps, a category of people (viz., the rich) who are generally more resistant to his message than another category of people (viz., the poor).

This makes eminently good, intuitive sense, too. Now I'm not suggesting Jesus was in some way showing favoritism to the poor and dissing the rich. No. He did realize, however, that the rich tend to be (or find it easier to be) more self-sufficient, independent, and proud than people who are just barely getting by. For illustrative purposes, then, he referred more to the rich than the poor in his parables. By the same token, however, I am not suggesting Jesus was saying there is anything inherently good or bad in being either rich or poor; rather, he simply identified himself more with the underdog than the overlords!

"For you know the grace of our Lord Jesus Christ, that though he was rich, yet for your sakes he became poor, so that you through his poverty might become rich" (2 Corinthians 8:9).

While Jesus evidently had a home in Capernaum for at least part of his public ministry, there came a time when his ministry required him to be homeless and dependent on the support of friends and benefactors.

"Jesus replied, 'Foxes have holes and birds of the air have nests, but the Son of Man has no place to lay his head'" (Matthew 8:20; Luke 9:58).

"Many women were there, watching from a distance. They had followed Jesus from Galilee to care for his needs" (Matthew 27:55; cf. Mark 15:40).

"As Jesus and his disciples were on their way, he came to a village [i.e., Bethany]where a woman named Martha opened her home to him." (Luke 10:38).

James, Jesus' half-brother, talked quite a bit about the rich and poor in his letter, and he pronounces a blessing on those who are poor in the world's eyes, the humble, the lowly, those of low degree, and those who do the backbreaking work of laborers but whose wages the rich hold back. See 1:9-11; 2:1-6; 3:13-16; and 5:1-6, for some interesting insights into the rich-poor dichotomy in James's thinking.

On the other hand, James denounces the rich with some harsh words (see 4:13 ff., and 5:1 ff.), and he even scolds believers for showing favoritism to the well-heeled, well-dressed, and gold-ringed, but contempt for the down-in-the-heel,shabbily dressed, and presumably ring-less (2:1 ff.).

In conclusion, Jesus may just have chosen to use more rich folks than poor to populate his parables as a way to underscore the inherently greater difficulty the rich have to surrender their independence and self-sufficiency to God. Whether rich or poor, however, we all need to become poor in spirit if we truly desire to inherit the kingdom of heaven and all the true riches found therein (Matthew 5:3; 6:20).

One consistent theme of the Gospels is the superiority of "spiritual wealth" (meaning the "richness* of a life lived in connection with God) over material wealth. This theme is exemplified in the Parable of the Rich Fool, Luke 12:16-21, and in Matthew 6:19-21:

Do not lay up for yourselves treasures on earth, where moth and rust destroy and where thieves break in and steal, but lay up for yourselves treasures in heaven, where neither moth nor rust destroys and where thieves do not break in and steal. For where your treasure is, there your heart will be also.

If you read with this in mind, the seeming paradox disappears. In several places in the Gospels the unrighteous rich appear, either in parables, or in actual interactions with Jesus. The theme of these interactions is nearly always that Jesus warns the rich person against his attachment to material wealth.

In many other parables, however, a figure of a righteous rich person appears (the father in the Prodigal Son, Luke 15:11-32, for example) --that person is generally a metaphorical representation of God, and the wealth described is spiritual wealth. Similarly, there are a few places where Jesus seems to praise someone who is seeking wealth (the dealer of pearls, for example Matthew 13: 44-46). In these cases, wealth is again a metaphor for the spiritual riches of a relationship with God.

This leaves the question of why money and wealth are so often used by Jesus as metaphors for spiritual righteousness. I would venture that it was aspirational for his audience. Many of his followers were desperately poor and probably coveted lives of wealth and ease. By framing the desire for righteousness as parallel but superior to the desire for material wealth, the parables make an abstract concept more visceral and easily grasped.

tl;dr version: Wealth is often used by Jesus as a metaphor for "spiritual riches".

To me it appears as though you're insisting on a critical scholarly approach to an elementary subject. The scriptures themselves seem to answer your original question of why there are more sayings about rich people than poor people:

"But they that will be rich fall into temptation and a snare, and into many foolish and hurtful lusts, which drown men in destruction and perdition. For the love of money is the root of all evil: which while some coveted after, they have erred from the faith, and pierced themselves through with many sorrows." (1 Tim 6:9-10)

The poor and meek were typically more righteous because of their humble circumstances.

"As sorrowful, yet alway rejoicing; as poor, yet making many rich; as having nothing, and yet possessing all things." (2 Corinthians 6:10)

Christ came to call the sinners to repentance, and rich people were apparently more inclined to sin, therefore, there are more analogies that relate to the rich, because it was the rich that needed to be called to repentance.

"... I came not to call the righteous, but sinners to repentance." (Mark 2:17)

It is possible that the answer to your question is, "No, this is not something that schools and scholars have addressed." You may be over analyising things with your research and reading too much into the subject.

The question as posed contains an implicit assumption that needs to be challenged -- or at least teased out into the open. It is this: that numerical represention of character types in Jesus' parables ought to reflect the proportion of attention Jesus gave to them outside the parables.

This is important for the the particular case of "wealthy vs. poor" characters in the parables, since parables "about" the wealthy typically warn against wealth, and so fall into line with what one sees elsewhere in the gospels concerning Jesus and poverty. In other words, it is not sufficient to do the counts/tabulations: one must take into account the intention of the parable itself. This point grows out of a relevant study by Stephen Wright, "Parables on Poverty and Riches", in The Challenge of Jesus' Parables, ed. by R.N. Longenecker (Eerdmans, 2000), pp. 217-239.

It should further be noted that with one solitary exception (as far as I know), the characters in Jesus parables do not appear as "individuals", but as representative types: this kind of farmer, that kind of despot, Joe Samaritan, etc. Again, this has signficance for OP's interests. W.R. Herzog makes this observation and draws out some of its significance in his Parables as Subversive Speech: Jesus as Pedagogue of the Oppressed (Westminster/John Knox, 1994):

To make sense of the parts, one must have some conception of the systematic whole. It does little good to recreate life in a peasant village unless one understands how the peasant village fit [sic] into the larger social, political, and economic scheme of things. ... These encounters [between peasants in the parables] are interpreted through "typificatory schemes"... Because the parables are full of typifications, their seemingly unique scenes and individual characters actually imply a social construction of reality in which people interact in typical ways. Even the characters found in the parables are not individuals but socially recognizable types who stand for larger social groups. (pp. 53-54)

The chapter that follows in Herzog's book explores and develops that wider social network.

In addition, assessing the balance of wealthy/poor characters in Jesus' parables should also take into account these two factors:

1. Of the thirty-seven parables in the gospels, I reckon about a third, maybe more, involve characters that are poor, maybe a slightly higher proportion if those "parables" that involve no human protagonists are excluded. So at least they're not absent. Here, too, as Herzog's observation suggests, recognition of characters embedded within the whole social scale is to be expected, not the isolation of any particular group. Thus we have both employers (wealthy) and day workers (poor) (Matthew 20:1-16), a rich man and a beggar (Luke 16:19-31 -- which is the case of the one "named" character in the parables), etc.
2. In many rural parts of the majority world today, subsistence farming is a way of life, much as it would have been for many of Jesus' listeners. What we sometimes fail to appreciate is that there are degrees of destitution. Some of the figures in the parables are hard to classify in these gradations of poverty. Given this social setting, Jesus' hearers from a broad range along the poverty/wealth scale of the time would have been able to recognize themselves in his stories -- even though they themselves might not have been "wealthy" by any measure we would recognize.

This all fits into the wider biblical witness. A certain distrust of wealth on the one hand, and God's "option for the poor"¹ on the other, runs through the Christian Bible -- Old and New Testaments alike. The challenging presence of the theme inspired the early monastic movement, with its embrace of "poverty, chastity, obedience", so it is no new idea for the Christian church.

For that broader context, an Old Testament perspective is provided by a recent book by David Baker, Tight Fists Or Open Hands?: Wealth and Poverty in Old Testament Law (Eerdmans, 2009). The study by Ronald J. Sider, Rich Christians in an Age of Hunger (IVP, 1977; later edition Nelson, 2005) has had prolonged influence, and brings the story into the New Testament.²

But the literature on this is, in fact, vast, and runs pretty much throughout the two millennia of the church's existence.³

Notes

1. As it has come to be known in some circles, associated especially with liberation theology and Gustavo Gutiérrez, although used much more widely now.
2. Original sub-title: "A Biblical Study", but in the latest edition: "Moving from Affluence to Generosity".
3. See also the literature cited in the Q&A on poverty in Luke's gospel.