It seems to me, instead of funding a new college or traditional research institute, some benefactor ought to fund a "research reproduction institute", dedicated to identifying and reproducing suspicious publications.

Yes, but in the vast majority of cases, it's hard to tell just by reading a paper if there's been dishonesty somewhere in the pipeline.

Also, the LK-99 example is an exception, not the norm–the chances of receiving significant attention for a replication study are near zero in almost all other cases.

I just don't think it's really relevant. If the research is impactful, then it'll be replicated (at least in part) when the next person tries to build on the results. If it doesn't replicate then they'll probably end up discovering something new/different - and that'll lead to it's own paper.

Even in the ideal world, you effectively almost never end up with a replication paper. Either it replicates and you add on your own novel research. Or it doesn't replicate and you discover something new

You can in theory end up with a super dull null result that disproves someone else's claim. But even then, when you set out on the project you're aim is to add something new on top of what's been already done. This happens all the time