There are huge problems in the world that desperately need solving. But most of the people who need those problems solved are too poor to pay for a solution. And most of the solutions depend on a major increase in knowledge (e.g. scientific research) which is very cumbersome to fund via a free market.
There's a huge class of huge problems that have known solutions, but neither the will or competence to implement them. It's also not a matter of wealth since developed world societies have nailed down a lot of problems despite starting at deeper levels of poverty.
it's not clear that's any better than just having the government fund the work directly.
Sure, it is. Government is absolutely shit at figuring out what is good research. One thing we need to remember here is that there used to be a huge, privately funded science powerhouse in the developed world. That got scrapped because it was easier and more profitable to siphon public funds than to do work that had actual risk to it.
In one possible future, it would be easy to find meaningful work solving the world's big problems but most jobs would be in the public sector - and taxes, on the rich at least, would be very high.
Not really. Welcome to the world of perverse incentives. Your bureaucracy goes away, if you actually solve the problem your bureaucracy was set up to solve.
In another possible future, the big problems wouldn't get solved and most people would be reduced to performing frivolous little chores for a small number of extremely powerful rich families in order to avoid starvation - but rich families would live out fabulous lives of idle luxury.
I think this is the actual future your ideas steer us towards. But fortunately, I have another solution. How about we just get out of the way of the people trying to work and the people trying to hire?