We did it for 4 years - packed our jobs in, rented our house out and moved to a property we already owned in Greece.
This was pre-Brexit, however - but I will stress that we still needed to pay for private health insurance over there (state pensioners were entitled to use the Greek healthcare system, we were not). It will cost several thousand a year. We also had to pay tax on our UK income - for the house rental - in the UK.
Obviously, we were not paying NI in the UK for those years so that was 4 years off our state pensions. We were able to make these up when we returned.
We lived on savings plus income from renting out our house and when we sold the Greek property, had made a decent profit, which put back the savings - that wouldn't happen now, I don't think.
What I will say are the downsides - apart from those already mentioned- are the ferocious heat for months, which effectively keeps you inside running the air con! If you don't speak the language really well, navigating systems (even with private healthcare - which we DID need and use - using it successfully was hard) can be quite stressful. Just paying your taxes in Greece is very complicated and we were advised to have an accountant.
Social services, etc in different EU countries and what they will do for you as you become elderly different widely.
You need to have quite a bit of money coming in regularly to live in most EU countries these days and/or own a property above a specific value. Each country has different residence rules - France, for example, will only let you apply for residence for up to 5 years at a time.