Firstly, the problem here is not an epidemic of scientists who feel too financially insecure to do good work. Many of the worst papers are being written by people with decades-long careers and who lead large labs. Their funding is very secure. They are doing bad work anyway for other reasons, sometimes political or ideological, more often because doing bad work results in attention, praise and power. Or sometimes because they don't know how to explain their chosen question, but don't want to admit that scientifically they failed and don't know where to go next.
Secondly, as you already realized your proposal relies on identifying which scientists have a proven track record, but the whole problem is that science is flooded with fraudulent/garbage claims which are highly cited ("proven") and which were written by large teams of supposedly respectable scientists at supposedly respectable institutions. Any metric you can invent to decide who or what has a proven track record is going to be circular in this regard. To Rumsfeld the problem, we are surrounded by "unknown knowns". You say this is an open question but to me that's a fatal flaw.
So the problem is actually the inverse. You say at the end, well, scientists who can fund their own work are an exception. Obviously in most cases scientists don't need to do this, they can also be funded by companies. Most computer science research works this way. Better CPUs and hardware is done almost entirely by companies. AI research has been driven by corporate scientists, and so on. In contrast academic funding comes primarily from government agencies that distribute money according to the desires of academics. This means a tiny number of people control large sums of money, and they are accountable to nobody except themselves. There are no systems or controls on academic behavior except peer review, which is largely useless because the peers are doing the same bad things as everyone else.
Viewed from an economic perspective academia is a planned reputation economy. The state is the source of all resource allocation decisions (academics being effectively state employees in most fields). There's also a deeply embedded Marxist worldview: universities have no working mechanisms to detect fraud, because of an implicit assumption that deep down when market forces are gone everyone is automatically honest and good. The hierarchy is stagnant; the same institutions remain at the top for centuries. A good reputation lets them select the people with the reputation for being smart (e.g. by school grade), so that reputation accrues to the institutions, which lets them keep selecting intake by reputation and so on. Supposedly Oxford and Cambridge are the best UK universities, they always have been, and they always will be. In a competitive, free market economy they would face competition and other institutions would seek to figure out what their secret is and copy it, like how so many companies try to copy the Toyota Way. In science this doesn't happen because there's nothing to copy: these institutions aren't actually different.
This implies a simple solution, just privatize it all. It would be wrenching, just like it was when the USSR transitioned to a market economy, just like it was when China (sort of) did the same. But one thing the 20th century teaches us is that you can't really fix the problems of a planned economy by tinkering with small reforms at the edges. The Soviets weren't able to fix their culture with glasnost and perestroika. They eventually had to give up on the whole thing. Replacing the current reputation economy with a real economy, with all the mechanisms that economic system has evolved (markets, prices, regulators, court cases, fraud laws etc), seems like a more direct and obvious approach to making things better, even if it may sound extreme.