Language is and always will be culture. It does shape how people think and act; for instance, when Eliza tries to return to the vernacular of her flower-girl days, she is unable--she has changed. Yet, she is not a lady, either, simply because she can speak like one. For, a social class entails customs, ways of thinking and acting that are inculcated through generations, not just months.
Perhaps we could, but why would we? Language is already, to some degree anyway, a divider of men (and women, of course). How one speaks impacts all kinds of things: how one is perceived, how well one is understood, one's self-confidence, employment opportunities, and more. The other trappings of upper class (clothing, manners, etc.) have the same kinds of impact. However, we all know that those things aren't what really matter.
More importantly, things like these don't make a person decent, kind, unselfish, or any of the other things which we most appreciate and admire in the people we want to associate ourselves with on a regular basis. Pygmalion shows us that in the persons of Higgins and Eliza.
It's what's happening on the inside that matters, and creating any elite category of people would most assuredly create attitudes and behaviors we would not want anyone to emulate.
Interesting question! I'm not sure why we'd want to build a temporary upper class society at all - Shaw's play "Pygmalion" seems to me infinitely more interested in how the way someone speaks can transform them, and the social value of accents and elocution. But I suppose that the play suggests that you could build an upper-class society based on Higgins' methods.
Liza says in the final scene to Colonel Pickering:
I can't go back to it. Last night, when I was wandering about, a girl spoke to me; and I tried to get back into the old way with her; but it was no use. You told me, you know, that when a child is brought to a foreign country, it picks up the language in a few weeks, and forgets its own. Well, I am a child in your country. I have forgotten my own language, and can speak nothing but yours.
In some magical, bizarre way, speaking upper-class, elocuted English has transformed the Eliza who was there before: somethign has been lost - it's not simply that she has learnt another voice which sits alongside her original one. It is like a tape which has been overwritten with a new song.
What, then, is Eliza - upper-class or lower-class? She herself argues both when speaking to Pickering:
You see, really and truly, apart from the things anyone can pick up (the dressing and the proper way of speaking, and so on), the difference between a lady and a flower girl is not how she behaves, but how shes treated. I shall always be a flower girl to Professor Higgins, because he always treats me as a flower girl, and always will; but I know I can be a lady to you, because you always treat me as a lady, and always will.
Does it depend on how she is treated? Certainly it is true that class roles depend very much upon outside viewing: so Eliza's "status" depends on precisely who is looking at her and judging. Yet it is also true that, even though she is a flower girl at heart, if she changes her accent, society as a whole will look at her in a different way - changing external factors in some way can perform a social transformation.
Eliza eventually argues that being a lady depends on more than learning an accent, but on internal behaviour and goodness - though even that is manifest in outward appearance:
It's not because you paid for my dresses... But it was from you that I learnt really nice manners; and that is what makes one a lady, isn't it?
Isn't it? Well, who knows. What does make you a lady? Is Eliza actually still a lady as the final curtain comes down? That decision, I think, could be differently played in different productions. Yet it is certainly true that society thinks she is a lady.
So the answer to your question is that we could certainly build something that looks very much like a new upper-class by training people in elocution and manners, and giving them new clothes and habits. But would we really be transforming them internally - or would it not be a lasting transformation? That, I think, is the real question of the play.