In particular, binary search on V, V >= time*money -> V/time >= money then dsu all the edges for which this is true. If the final graph is connected then V can be reached if not V cannot.

This solution seems oddly fast considering the constraint is M <= 10,000, so what is the mistake here?

Additionally, what is the intended solution with MST?

You are right it is not correct. For some reason I thought that if all the edges satisfies V/time >= money then the entire graph satisfies this property, but this is definitely not correct. What is the intended solution though?