What was Buddha's purpose in constructing The Four Stages of Enlightenment?

In the most simplest terms,

1. Stream-enterer (Sotapanna) - you let go of the belief that you’re really a separate self living inside your head and looking through your eyes. This illusion is eliminated, and when you look within you can’t find a self anywhere.

   Stream-enterers have also let go of attachment to rites and rituals and doubts about the teachings.

2. Once-returner (Sakadagami) - returners have entered a stage where their “no-self” is completely integrated, and they experience a significant reduction in attachment and aversion and the suffering that accompanies these states of mind.

   They are much more detached. For example, occasional irritation and preference replace hatred and greed, which no longer have any hold over the once-returner.

3. Non-returner (Anāgāmi) - The Non-returners are free from feelings of sensual desire and ill will towards others. They are seen to be partially enlightened and on the way to complete enlightenment.

   They experience significant levels of compassion as they have lost the sense of separation between themselves and others.

4. Arahant - The fourth stage is free from having any craving for prosperity in the material world, doesn’t crave existence in the ideal world (which is heaven), doesn’t experience conceit, never feels restlessness and isn’t subject to ignorance.

   At this stage, the person is experiencing ultimate nirvana. The feeling has been compared to falling into the depths of a cloud and disappearing. The circumstances of life no longer stir even the slightest craving or dissatisfaction.

Link :- The 4 Stages of Enlightenment According to Buddhist Scripture (And How You Can Achieve Them)

The path to attain Nirvana is The Noble Eight Fold Path.

I think the other two fetters, that define a Sotapanna, are no less significant than "seeing through ones own imagination of their self" -- i.e. they are:

• vicikicchā (doubt, uncertainty) ¹
• sīlabbata-parāmāsa (attaching to ceremonies)

Maybe these are a superset of (more general than, thus greater than) the "overcoming sakkāya-diṭṭhi" attainment.

Also I think that sakkāya-diṭṭhi is more and less than "one's own imagination of their self" -- I think it's literally "true-body view", and its antonym is to not seeing (or view) the five aggregates as self -- which I think is an intellectual position or realisation and, moreover, an experience.

It's a preliminary attainment, see e g. How are 'conceit' and 'identity-view' not the same? (or here where I describe that as analogous to being an addict in recovery, still experiencing conceit and suffering and so on).

I gather that a further thing that's required of a Sotapanna is heedfulness, i.e. though they may have heard (and understood) what right view is, but it takes heedfulness to keep it in mind (see e.g. Jayarava's analysis of The last words of the Buddha for a description of appamādena).

Also I think the doctrine (of "four stages") meets the criteria defined in the handful of leaves, i.e.

And why have I taught these things? Because they are connected with the goal, relate to the rudiments of the holy life, and lead to disenchantment, to dispassion, to cessation, to calm, to direct knowledge, to self-awakening, to Unbinding. This is why I have taught them.

However, what practical use can one obtain from using the four stages?

Yes I think the suttas say more (or in more detail) about the right conditions for stream-entry than later stages (see also Can someone give references of the four stages of enlightenment in the four major Nikayas?).

I think the other stages (or other fetters) is a reminder of how much more (or further) the way to cessation is than only, merely, or even wholeheartedly adopting "right" view (intellectually) -- i.e. after that there's still overcoming any anger, sensuality, clinging, conceit, and so on -- see also e.g. What is effluent?

¹ Looking for definitions of vicikicchā, I found vicikicchā and suttas that define it, which says (I summarise):

• AN 1.15 and 1.20 says doubt arises and increases with unwise attention (and, vice versa, decreases)
• DN 2, DN 25, and MN 39 suggest it means "uncertainty as to what things are wholesome"

And:

• At MN 16, vicikiccha applies to Buddha, Dhamma (that includes any part of the teaching, the noble truths, the three characteristics etc. and also who speaks the truth or falsehood [SN 43.13 and AN 3.66]), Sangha, the training (sikkha).
• MN 23, it is compared to a fork
• SN 22.84, doubt is compared to a "forked road"
• SN 12.32, doubt is about whether one as abandoned the asavas, in other words whether one is an arahant or not
• SN 46.51, doubt is twofold: there is doubt about the internal and doubt about the external
• AN 7.54, doubt does not arise about undeclared things because of the cessation of views

I wondered about the definition in SN 12.32 -- if overcoming doubt is associated with being a sotappana, why is it still being mentioned in the context of deciding whether one is an arahant?

I think the answer to that is here, part of the definition of "conceit" ...

a Sotapanna can only give rise to true conceit, viz:

1. thinking one is superior when one is superior
2. thinking one is equal when one is equal
3. thinking one is lesser when one is lesser

... i.e. a Sotapanna, when conceited, would see correctly that they are not an arahant.

Actually it's not constructed by Lord Buddha, it just nature.

The four stages define what the basis of enlightenment is, namely, losing the belief in an inherent self ("stream-entry") ; and, at the same time, further define what full enlightenment is ("arahant"), namely, the complete uprooting of greed, hatred & delusion. Therefore, if you or a teacher (guru) still has lust, hatred or selfishness then this means full enlightenment has not been reached.

For example, there are many intellectuals (e.g. Sam Harris) who speak and act to give an impression they are enlightened because they speak theories about "no self". However, if we examine their other views and beliefs, we can work out if they are really enlightened or not.

Or there is a certain chat site where members claim to practise "jhana" yet they also post about their sexual lusts. Since a master of jhana will be at least a once-returner but often a non-returner, any significant lust should not arise in a once-returner or non-returner.

The same applies to us. The four stages are a guide to us of the level of (our) enlightenment. They are not for boasting but for the opposite. They are so we do not overestimate our experience.