Should a Monk speak out in favor of doing good?

Not sure of what Mr. Ryan speaks (reading just the word), but Atma can remember that he once had similar thoughts in regard of Buddha. Thinking he might had caused a lot of people suffering by being attached to him.

But the more one knows the real teachings, one sees that he actually did not let such a kind of kamma behind. Its simply by people taking it wrong, that they have problems.

There is of course this danger, which you mention, that one would automatically accumulate everything, starting form the moment people see "ohh that is not a normal one" but there is nothing better as to practice the 13 Dhutangas which protect one self and people well and actually the most serious monks keep them.

Taken for example SEAsia. If people get to know that there is anywhere a Dhutanga Monk, they would come form near and far and would like to make donations, since such is a high merit. If people (monks) do not really stick hard to the Dhutangas they are caught immediately. One sees that well when observing the Thai Forest Tradition today. Their honor had actually "killed" them practical and they have already become "normal" ordinary people. Yet still living form the reputations of their teachers of old times.

So there are certain dangers and they are even covered by a downfall rule, for hard cases:

Should any bhikkhu, without direct knowledge, claim a superior human state, a truly noble distinction of knowledge and vision, as present in himself, saying, "Thus do I know; thus do I see," such that regardless of whether or not he is cross-examined on a later occasion, he — being remorseful and desirous of purification — might say, "Friends, not knowing, I said I know; not seeing, I said I see — vainly, falsely, idly," unless it was from over-estimation, he also is defeated and no longer in affiliation. BMC1

But as you can read there, even if people out of faith would address one Arahat, there is no fault for one. So if people "love" one, one having practiced proper, does not make anything wrong. That's great, even mostly not welcome for the serious practicing monk.

There have been occasions where when lot of people came to visit the Buddha and he denied. Even if monks begged him "Now is the time for the Blessed One's acquiescence, lord! Wherever the Blessed One will go now, the brahmans of the towns & countryside will be so inclined." he refused, at least saying "But when I am traveling along a road and see no one in front or behind me, at that time I have my ease, even when urinating & defecating." Yasa Sutta: Honor

So in regard of your formulation: No, there is no reason why a Bhikkhu should step back form his virtues, what ever they are or to even cheat, so that he would prevent certain attachments and problems with fans and so on.

If one reads the Suttas, one could easily believe that its all about self-aggrandizement, but actually it was a certain scarify of the Buddha, to that extend, that even no monk after him would need to use it in the same way he did. (Atma guesses the questioner understands what he tries to say)

One does actually not easy accumulate when always straight and honest (in a sense of beneficial and true for the path) and without any flavor of socializing or concern about reputations.

As for the OP question it self: "Should a Monk speak out in favor of doing good?" such is most critical, if speaking what people like and mostly a heavy offense which is called a corrupter of families, sadly one of the most, even whole communities faults.

This is illustrated in the origin story of this rule, in which the followers of Assaji and Punabbasu (leaders of one faction of the group of six) had thoroughly corrupted the lay people at Kīṭāgiri. "Now at that time a certain bhikkhu, having finished his Rains-residence among the people of Kāsi and on his way to Sāvatthī to see the Blessed One, arrived at Kīṭāgiri. Dressing (§) early in the morning, taking his bowl and (outer) robe, he entered Kīṭāgiri for alms: gracious in the way he approached and departed, looked forward and behind, drew in and stretched out (his arm); his eyes downcast, his every movement consummate. People seeing him said, 'Who is this weakest of weaklings, this dullest of dullards, this most snobbish of snobs? Who, if this one approached (§), would even give him alms? Our masters, the followers of Assaji and Punabbasu, are compliant, genial, pleasing in conversation. They are the first to smile, saying, "Come, you are welcome." They are not snobbish. They are approachable. They are the first to speak. They are the ones to whom alms should be given.'"

Abhaya Sutta: To Prince Abhaya (On Right Speech) shows well that sweet talks are nothing that are welcome if it is not simply for the path and to let go.

So it should cover all possible aspects of the question and at least maybe, a monk is no and should not be a social worker. That's a job he left behind by going forth. (females are always included speaking of monk)

Maybe one in addition. It is not allowed (out of this reasons) to receive a donation after having taught the Dhamma. So if one likes to benefit a monk, he would do good to do such in advanced. The story about the Buddha in such cases can be read here: Discourse to Bharadvaja, the Farmer A good talk about the matter of being first on giving is: Better to Give than to Consume, if one like to understand the matter also from a practitioner side as a layperson.

So there are certain ways to cut off the danger of doing a favor but less really less make such their habit today. Its all about gain, becoming, honor and for "higher purposes" (for simply this world) and seldom for Nibbana.

Asked: "others may see this motive as being self-serving; and therefore how to address this properly?"

One somebody may think, one can not manipulate or change and there is also no use for such (if one is not attached to his reputations). As seeing the motive and virtue of a monk and if he is walking the path is very difficult to find out for laypeople. Actually only a one already on the stream would recognize a straight person (in Dhamma sense). So once a king tried to test the Buddha and taught him about the possibilities to judge a person: Paṭisalla Sutta: Seclusion and it end (after an explaining with a important exclamation:

One should not make an effort everywhere,
should not be another's hireling,
should not live dependent on another,
should not go about as a trader in the Dhamma.

And then, taking the original question again:

If a monk is given the opportunity to speak in favor of doing a good deed that a supporter of his wants to do, even if that good deed directly benefits him, is it proper to do so?

Let Atma bring a sample of how such could look like. One likes to give a donation and would ask "do you like coffee/or water?", than he would stay silent. So suggestions of that kind are not proper. If one actually gives, fine, but usual ways monks on many places even tell people "you have to ask me first" (as the other extreme) is simply wrong. It also happens that monks would say "If you give me a donation, you make much merits" or other things. All this kinds of "doing others a favor" are fine, even it would be such (which not even the Buddha told people, given that they would have made much merits if done so)

So even hinting is walking on the edge. Good teachings about that are this short essays: No Strings Attached, The Economy of Gifts

(Note: this answer has not been given with the agreement to be means of trade or the purpose of/for trade and/or keep people trapped and bound. How you handle it lies in your sphere, but does not excuse the deed here either.)