The real value is in advancing the understanding but very few outside academia understand this (it’s counterintuitive, which is fair enough as it’s somewhat unique to mathematics as a subject), so if you want to get funding you frame it as being for solving the problems.

Now that you can solve without understanding this setup is broken. Either the sources of funding will finally have to learn the difference and the value of the latter, or mathematics research ends.

That's not quite right, assuming everything said so far is just flat true.

The funding can have been for advancing the understanding, by using a more measurable proxy and reasonable target that closely aligned with advancing understanding.

Perhaps another phrasing might be "we are paying people to go through the process of solving these problems" rather than "we are paying people for solutions". I might set a random task for my kids while on a hike to find X things, not because I want to find ten different leaves but the process of doing it means exploring and investigating in a certain kind of way a certain kind of area. If some sets up a leaf selling stand, they have advanced the field of "finding leaves" and kids can now very very easily get ten different leaves.

Now leaves here are frivolous and not useful. That example works better looking at, say, homework - clearly I don't care about having a list of words spelled correctly and I don't need the answer to 5x7, nor do we need more book reports on The Great Gatsby. We're doing them to teach, it's very explicitly about the result.

Research level maths however is a bit of both. The answers to some of these things are genuinely useful. Having the answer may be better than not having it. But having a lot of people working on solving it has other useful and beneficial outcomes.

We have structured large scale systems of huge numbers of people and institutes around how this works, and what top mathematicians are telling us is that open problems (particularly at new researcher level) are a key part of this process and are hard to find. Academia changes incredibly fucking slowly, just glacially slowly. Some aspects (most?) are barely changed across hundreds of years. And across an incredibly short space of time (less time than one paper can take to go from fully finished to actually published) we have gone from "this machine can solve school level work" to "this machine is solving major research level maths problems". The existing system will not work, the machines will not get dumber or slower, and some of the impacts are things you cannot undo.

Wouldn’t be the first time someone kept the bean counters happy while also generating some positive externalities! Of course, now with AI we might finally be able to realise maximum efficiency, where only the bean counters are happy and there’s no other positives.

No, that's demonstrably untrue given how little direct value is given to most of these results.