Choosing Beggars

Those who tip well generally make up for those who don't. And those who don't tip DO pay the restaurant.

What would happen if tips were removed and prices adjusted so the price for everyone was the average of bill+tips?

The cheap bastards who don't tip right now but keep the business afloat would stop going out because it's "too expensive," and those who tip well would be paying a lower average price, so there would end up being less money going to the restaurant AND the staff, while all the fixed expenses would stay the same.

Reality is the current shitty system actually maximizes income for waitstaff.

Waiting tables is one of the few career paths where someone can make a living wage without any special certifications, licenses, or education. Tips are why that works.