One existing institution that is fairly close to you're imagining is the HHMI.
However, I think you are making two assumptions about research that are incorrect.
First, it isn't at all clear that separating teaching from research entirely would be a good thing; researchers need to be trained, and trainees bring diverse skills and perspectives to the labs in which they work.
Second, many of the "smaller problems" have been getting bigger, in that they depend on large expenditures for equipment and professional staff to keep things running. To execute your example of characterizing digestive flora from different countries, you'll need a facility with the latest sequencing equipment and people to run it, computational infrastructure, and boots on the ground to collect the data in all those developing countries.. it'll quickly turn into a multi-million dollar collaborative project.
Yes, I think the funding is the key. There are already funders who see it as their mission to support scientists and allow them to tackle interesting questions. But in the end, they have a limited pot of money to dish out, and they have to somehow make a decision about what's worth funding. That brings us back to spending a lot of time writing grant applications and trying to justify the work.
Ultimately, 'gentleman scientists' were probably able to follow their interests because they were rich enough that they didn't need to worry about earning a daily wage. That's still possible, but we'd much rather let science be done by anyone who's interested and good at it.
The OP mentions Microryza, their crowdfunding platform for science. That's a very interesting approach, but I think it only works if you already have some kind of paid job in science: the projects I've looked at are all asking for much less than a year's salary for one person.
However, I think you are making two assumptions about research that are incorrect.
First, it isn't at all clear that separating teaching from research entirely would be a good thing; researchers need to be trained, and trainees bring diverse skills and perspectives to the labs in which they work.
Second, many of the "smaller problems" have been getting bigger, in that they depend on large expenditures for equipment and professional staff to keep things running. To execute your example of characterizing digestive flora from different countries, you'll need a facility with the latest sequencing equipment and people to run it, computational infrastructure, and boots on the ground to collect the data in all those developing countries.. it'll quickly turn into a multi-million dollar collaborative project.