Well if you're going to take things absolutely literally the alternatives aren't so great. They are all products of the fact that we're paranoid about giving orders to people.
"I would like a cup of tea" is just a statement of fact, not a request. It leaves it up to the other party to decide what to do about your desire. The response might be "Would you indeed? So what's stopping you from asking for one?"
"I'll have a cup of tea" is more like a prediction. If all goes well, in five minutes I will have a cup of tea. We still don't know how this is going to come about.
"May I have a cup of tea" is speculating on the possibility of having a cup of tea some time in future. The answer might be "Yes, you may, but first you must order it".
"Can I get a cup of tea" is not much better. It's asking whether it's possible to obtain a cup of tea. It's a bit redundant though, especially if there is a sign above the counter advertising tea. Of course you can get tea here. It's a tea shop.
But all these phrases are dancing around the fact that we are giving an instruction to the person serving. Why not just "Give me a cup of tea!", or if we are feeling polite "Please give me a cup of tea".
Best to just be brief. "A cup of tea, please." OK, it's not a proper sentence as it lacks a verb, but it's unambiguous, to the point, and has "please" on the end, so it's polite.
Of course, they all work. They all get the message across, which is what language is for.