I doubt Americans will even pick up the phone or respond to LinkedIn messages / emails when they will se the budgets for the software Engineering roles in the EU.

I am saying that as an European, just to be clear.

I know several folks who've migrated from US -> EU tech roles in the last few years. Yes, you earn less and pay (somewhat) more taxes. But if you have a few kids the difference in cost of education pretty much wipes out the difference, and some folks really value the stress reduction of a robust social safety net (layoff protections, healthcare coverage while unemployed, etc)

With a baby on the way, I'd seriously consider it for their lifetime benefits. Where does one begin looking?

Not everyone is optimizing for total comp. Some are optimizing for better lives. It's not a wild concept considering how many people get pulled into startups, 90% of which fail, under the guide of "mission" and lower market comp. Do you pick a mostly assured better quality of life? Or an equity payout lottery ticket/fairy tale? Certainly, there is a minority of folks making wild comp at FAANG, but that is a privileged minority of total tech and IT workers.