It's really easy to remain in EU forever as an illegal immigrant, in some places your immigration status will have no effect on your access to free healthcare.
Nobody has ever checked my immigration status while renting in the EU, nobody has ever cared about the fact that I work for an US company owned by me that pays my salary to my US account.
Sure, maybe you'll have a bad time renting in France. Don't go live in France unless you want to have a hard time while renting.
> A good friend of mine, without papers, was living 10+ people in a 2 bedroom flat
A good friend of mine, is here illegally and lives in a nice central 2 bedroom flat in Barcelona all by himself. He had zero trouble finding an apartment, the landlords just don't give a shit.
Many illegal immigrants are completely broke, but that's by no means a requirement.
It's a hard life in the US, often with access to even less things than illegal immigrants have here. Not sure what this "constant risk" is, unless you engage in violent crime you'll find it very difficult to get yourself deported.
By far the biggest thing you lose is the ability to fly around easily, but it remains (and will remain) easy to sneak out after overstaying your visa waiver.
> It's not "really easy" at all and I don't know why anyone would think that
Because I know multiple illegal immigrants who vastly improved their lives by moving. A good friend of mine spent 4 years going in and out of KS state prisons, after a while he decided that he needed a change of scenery and hopped on a plane to Spain.
There's little chance he could get a visa without lying, but that hasn't really affected his life. He makes a living freelancing online, gets paid to his US account and pays the IRS. Here he enjoys access to excellent healthcare without risking bankruptcy.
And I’m sure you will have access to universal health system and private ones.