The same could come to Europe (lowballing broke people), because you can just make employees give their consent by baking it into all employment contracts. Just like the working time directive opt-out (HR say "take it or leave it" to 99.9% of people).
It probably already happens where it's already acceptable to request financial checks such as the finance industry.
You could try, but that would most likely be illegal and also fought tooth and nail by unions, which aren't as neutered as American unions. In Europe, sympathy strikes aren't Verboten like in the US.
The problem is the party selling that data actually. The nurses would need to consent that whomever the data broker got that data from to share their data with the broker and for the broker to share that data with the nursing service.
I don't agree. Which laws? Putting it in the contract is giving your permission. You're allowed to give your permission under GDPR.
A similar thing happened with the Working Time Directive - almost all contracts in my country make you agree to opt-out and that's been like that for over 15 years
That's not how European courts have been interpreting it. When they say "free consent" they do mean free as in beer, so a "consent or loose your job" provisions would almost certainly be thrown out.