UEFA has been horrible in their dealing with the "bigger clubs"? Specifically? As in more terrible with them than with every other club?
Greed from UEFA is in the form of corruption from the people in charge of it, because otherwise UEFA is not a corporation designed for profit, it's an associative body to govern football. People in charge of those 6 British clubs, for example, are in it for straight profit and/or influence. These are significantly different types of greed, because one is a flaw of the system, which can ultimately be corrected, the other is just inherent to it, i.e., people who own the power in those clubs aren't really in it because they love football as a sport.
As an associative body, UEFA rules, and it's competitions line of evolution in the last decades, is incredibly biased towards maintaining status quo. But they still have to maintain a resemblance of being an associative body with same rules for all, etc. This completely breaks away from it in that core principle of 15 clubs who have their presence guaranteed.
And if we're honest, this is the only way to maximize profit in a straight-profit corporate world. Just don't fool yourself. These clubs aren't breaking away from UEFA to create a fairer and more functional sportive competition. If they were, they would have organized this in a completely different way, with completely different principles.