That wouldn't really help. Even if only publicly-funded lawyers could represent criminal defendants (which I don't think anyone actually wants), those with financial means would be able to employ an army of experts, researchers, investigators, etc. to aid in their case, in addition to the biases juries have toward those with means, and political sway the elite hold. And if you somehow prevented that inequality, you'd just drag everyone down to the lowest common denominator, and all the more people would be harmed by unjust prosecution.

Every criminal defendant (at least for felonies or serious misdemeanors) in the US has access to a government-funded lawyer, if they can't afford their own. There are over 15,000 public defenders in the US, who get a fixed salary as you mention. They are often overworked, and have limited time and resources to help the people they defend, but they help countless people every day. I'd like to improve the funding and the number of public defenders available (and of course decrease the number of people on trial for nonsense). I'm glad the system exists. It's helped friends and family, with largely positive outcomes that I don't think would have been any better with an expensive private lawyer. When it comes to more serious crimes, you probably don't want to be in that situation.