Wages for those will rise accordingly, benefitting local nurses (but raising costs for nursing)

You could have auctions per profession.

You could, but the whole point of the auctions is to allocate the visas economically efficiently, which this would undo.

No it would not undo it, it would allocate the visas economically efficient within professions, while also allowing the government to prioritize some professions based on concerns other then economic efficiency.