Travelling on HOLIDAY to various countries in absolutely no way, at all, represents the realities of living in any of them though. I've lived in a few EU countries. I've also holidayed in those same ones beforehand.
And there are so many other considerations. Tax being a biggie. How does the country you choose require tax by residents who earn from abroad to be paid? In some countries, NL for example, it's not so straightforward to live there and be paid from a company outside the country (more so for the company if it doesn't have an office in that country). Maybe your company is paying tax on your behalf and this isn't an issue, but most companies who do that don't have it as an indefinite offer. So ultimately, how much tax do you want to be paying? Presuming you're on an "expat" style income, in NL you could lose more than 50% to tax + medical + water taxes. Of course, you get things for that, but looking at your income and school fees really isn't the calculation to start off with.
Then there's the medical side. Do you speak any European languages? How much time are you willing to invest in learning one? Do you expect that you can get medical treatment in English? Do you expect that when you're living, not holidaying, that you'll be able to integrate locally without language? If not, then which language is going to be impossible (Hungarian and Finnish I'd start that list with), and which will be easier? Or will you really not want to learn another language, in which case Ireland and maybe Malta are places you should start the search.
Most people don't have the "luxury" of choosing anywhere. They have to deal with the country they are put in for work. But when you have the choice, there really are considerations to make.
And another one not to overlook is your marriage contract (assuming you're married - if not it's a whole other issue). Worst case scenario, would that country's family law support you moving back to the USA with the kids, if your marriage breaks down? This isn't a joke btw. I know plenty of foreign women (also with EU nationalities) stuck in EU countries because of divorce. They can't get a job, spousal maintenance isn't much/long and they're seriously struggling.
I'm not a Debbie Downer here. Plenty of people move to "Europe" and like it. But I've also met many, many others who are shocked by the difference with the US. And I've met, sadly, a great many mothers (in particular) who have not loved it and are stuck here. None of them had any idea of the possibility of what happened to them happening.
You really need to look into this a bit more and make an informed decision. Or you'd may as well save your energy and just stick a pin in the map and go to that country, then make it work, whether it's Latvia or Portugal. After all, no European country has really bad education and they all have private schools offering English language education.