Perhaps that's because you're trying to legislate emotion instead of an objective difference.
If the rules should apply to one type of organization and not another (your "higher purpose" opinion), then it should be straightforward to unambiguously describe the difference in law: a bright line test. If you cannot, it's likely you're just trying to encode your own subjective personal dictatorial preferences into the law, which is contrary to the values of a society that believes in the rule of law.
It's possible that you just don't like McDs, or do like AI. That's fine, but trying to shape tax law based on your own personal value judgements is contrary to the ideas of equal application of the law to all members of society. This is sort of the "I know it when I see it" approach, and it's, in my view, a somewhat dishonest way to try to legislate personal morality into the public sphere.
What taxes would AI pay? They're not selling items for profit. The only change that would need to happen is removing the ability to deduct charitable contributions from taxable income. I don't see any problem with that. People can still donate to AI with after tax income.
Not-for-profit firms make profits all of the time, and while I'm not familiar with UK tax codes, at least in the US the only fuzzy line is that pursuit of profits not be the end goal... quite a few "non-profits", such as hospitals, sell a large number of services for a profit!
As a soccer player, it might be a good move to fall to the ground when the ref cannot see you. But you are an asshole if you do it. The rules of the game are fine, the player is an asshole.
Maybe you should consider the hate of the public just as part of the game these companies are trying to play?