Considering that I've personally went and lived in other EU countries 3 times by now, you're way overcomplicating it.

Residence is only important for stuff like tax purposes (to avoid that people spend a week in a low tax country and claim they're resident there for tax evasion purposes) and to avoid Health Tourism in countries with a national health service (were people might otherwise come over for a week just to get free/better treatment).

Your right to live in an EU nation as an EU citizen are not affected by "residence status" as they would be if you're in a foreign country with a visa system because it's not the same kind of residence - it's for tax purposes not for employment or access to services as it would be in the US (or for a non-EU citizen in the EU).

That "residence document confirming your right to live there" is not required by anybody for providing you goods or services (they're not even allowed to make anything for EU citizens conditional on "residence status"). You can get the document and then the 3 months apply, but it's only ever needed when you're filling taxes in the country you were living in before to prove to them you're a resident elsewhere and hence will be paying taxes elsewhere (so, for example, when I left The Netherlands to go live in Britain I had to inform the authorities in The Netherlands that I was now resident in Britain for tax purposes, though I think I didn't even need to provide them any document as these systems are integrated across the EU).

Health insurance is only for some countries. In my home country - Portugal - we have a National Health Service so any EU citizen over here who is not a Portuguse can just use it for free like everybody else. Again any such 3 months rules would only apply (if this country actually applied it) to try and avoid Health Tourism and if you're an EU citizen just get a European Health Card (which is free) and you're covered by your home country even whilst abroad for those first 3 months (supposedly your country of residence is the one that covers the costs of your health treatment in another EU country). It's actually a good idea to get that card even if only going on vacations in another EU country since, for example, if you have an accident there and end up in Hospital you're covered by whatever healthcare arrangements you have in your country of residence.

Deportation is for things like one having murdered somebody in the host country, serving a sentence and then getting deported. It really has to be this extreme and is incredibly rare to happen. EU countries can't just deport EU citiziens without quite a heavy reason they can justify - it was actually part of the bitching and moaning of the Brexiters in Britain that they couldn't just deport EU Citizens.

The general rule in the EU at a treaty level is that citizens from other EU countries cannot be discriminated against compared to citizens of the host country. Yeah, you found the details related to avoiding that people evade tax by gaming the residence status for the purpose of paying less or no tax, do Health Tourism or just come over and start living of the Social Security in a rich country.

You absolutelly can just get on a plane and go stay on an EU country pretty much unprepared and then you have 3 months to figure out if you want to stay and only by then do you need to do stuff like register (only for some countries) and get health insurance (again, only for some countries: those where health insurance is mandatory by law for everybody). Further, you can get a job there on day 1, since you have the right to do so anywhere in the EU and your residence status is irrelevant (and in fact plenty of freelancers working in a Services domain will just go to some country, do some work there, and then come back and it's actually the intention of the EU treaties that they can do so: it's the Freedom Of Movement required for the Freedom Of Trade part to apply to Services, not just Products).

I've done it like that twice, first when moving to the UK (hopped on a plane, stayed in a hotel for 3 weeks whilst I looked for a job, ended up staying there for over a decade) and latter to Germany (where I left before the 3 months were up as I could to the work I was doing elsewhere with lower living costs and knew I would need to start paying health insurance in Germany after those 3 months).

I also have family members that do the Freelance thing of just going to another country in Europe, working there for a couple of months and then coming back.

Agree with you that people should be informed (which is why I knew when I went to Germany that I had 3 month to decide if I would stay or not and that if I did I would need to register and get Health Insurance), just disagree that is in any way a significant bump in the freedom to just go to another EU country to live and work there - the biggest bumps are cultural, linguistic and having the money to pay for a place to stay whilst you find your first job.

PS: Oh yeah, and taxes are also a bit of a bumb since you have to figure out which country you're supposed to filled them with and pay them to based on how long you've live where and most people really aren't used to it. If I remember it correctly in some cases you might have to fill and pay taxes in multiple countries depending on how long you worked in each during the tax year, plus different countries have different tax year ends, which adds to the mess.