With COVID-19, I was wondering what situation is on border controls. I haven't visited EU post COVID-19 (been to UK but not mainland Europe) so are border controls abolished or do they have them in place to control spread of COVID-19 such as checking people are vaccinated (and if not must self-isolate for 10 days) as well as requiring testing to cross. If COVID-19 doesn't end anytime soon, could this impact free movement? Yes right to live and work anywhere in EU won't be impacted other than those moving have to self isolate for first 10 days.
I think chances that you are "randomly checked" have increased. But very few countries perform border checks like during the first wave anymore. If you have a QR code you're good to go. If you don't, travelling to other countries by car would still be easy even if you technically break the law (on airplanes this isn't possible because they ask for your international QR); most of the time you wouldn't get caught. But because of the fact that most countries have 2G or 3G as domestic policy, you'd still not be able to book a hotel, eat in a restaurant, go to a theater etc. in your country of choice.
There are a few notable exceptions, though: some countries go further than the EU and have additional testing requirements (Portugal) or exclude certain vaccines (J&J; Austria). In my opinion, the latter is particularly bad practice: many people in the EU didn't get to "choose" their vaccine of preference (Dutch PM Rutte himself, for instance, was born in 1967 and therefore received J&J) and could never have known that countries would be going rogue again or exclude a vaccine promoted as safe everywhere just months before. The EU is a joke if it cannot ensure countries adhere to the same standards.