So more people learn that trade and another one loses its staff.

Labor shortages aren’t just money. Nauru has tech labor shortages. Singapore has notable farmer shortages. There are teacher shortages in the US even where wages pay well because teachers are sick of regulations or students. Korea has such an inverted population pyramid that its only possible future is half the country devoting their lives to caring for old people at the expense of either industries having no workers, and immigration isn’t a solution because other countries will be hitting similar problems at the same time.

Depending on the labor market in question, there could still be a labor shortage though. Those new nurses aren't just appearing from the aether. In places with very low unemployment they'd likely be leaving other jobs. You're then only moving the labor shortage to some other area of the economy.

Sure, now you might have enough ER nurses, but maybe now you've got a lack of primary care nurses. Or nurses at nursing facilities. Or maybe instead of just drawing from the pool of nurses you've now got fewer doctors or PA's, since you've now incentivized nursing over being an MD.

Sure, but what's the lag time on that? Four years? Ten?

It's possible to claim they should have paid more in the past, and that's part of the problem, but unless you're going to help immigrate already-qualified nurses from another country you can't simply increase wages and solve the problem you're having today.

You can kind of. You can raise the pay for nursing assistants and lower the credential bar for actual nursing, and raise pay to entice those that quit to come back. There are probably enough skilled folks out there to make that happen