I love the idea in theory, but in practice, I just see a load of work. For me, they're in the same category as beautiful, ornate, massive gardens: amazing to enjoy, but so, so much effort to maintain - and I'm not sure it's worth the hassle.
It could work if you bought it and then rented it out, hiring a local cleaner and maintenance people when needed - although you'd then have to pack all of your private stuff away into a big lockable cupboard every time and would have to plan when you wanted to go well in advance, to make sure you didn't accidentally have guests booked in at the same time.
I think very rich people make it work by retaining/employing staff in each house - they do nothing mundane themselves in their main home, so all a second home involves (for them) is the travelling (probably with a chauffeur/helicopter pilot).
Even then, they often just pay all of that money to own them, but never visit, purely for bragging rights. I remember reading years ago about an Arab sheikh who paid £100,000 a year to rent/lease a prime parking space in an underground car park in Knightsbridge, as well as a top of the range Rolls Royce to park in it. It was later discovered that he hadn't visited the UK in over 20 years. What's the betting that he had the exact same set-up in Paris, New York, Sydney, Tokyo, Rio etc. - but then, if you're so rich that doing so would cost you the equivalent of 20p a year to an 'ordinary' person, it's probably just a little bit of harmless fun and something to boast chat about at dinner parties, as far as they're concerned.