2 of 2 edited to emphasize safety

That would be an excellent way to start a fire, so yes, DO limit current to manufacturer recommended values.

But to answer the question: no downside. I prefer to charge up to 4.1v to make my cells live longer but that's unrelated.

The key point you make is: "so long as the maximum current is limited to a reasonable value for the cell (say 0.5C)" That alone will make it so the 4.2v you aim for will be dropped so as to push only 0.5C in the battery at a given instant.

My experience: I actually tried to push a constant 4.2v with NO current limit through a li ion cell.

The amps can reach double digits and the cell hates it so much that it heats up like crazy.