Absolutely you can. The places in France and Spain I've flown to just suggest you bring your own pedals, so they match your shoe cleats; they'll fit them before your hire. You can usually bring your own saddle too. It's far more convenient than bringing the bike.
I've also done it the other way, my main bike has S&S coupling so I could bring it aboard the Eurostar. For touring I prefer my own bike, because I have the racks set up for my panniers, but when I do that, I prefer travelling by ferry/train.
I love it! Even for the lift/shuttling-bike-park crowd, I bet many places would happily install cleat-matching pedals (or your own), your saddle... maybe your grips if you are picky?
Privileged enough to fly somewhere specifically to ride bicycles? C'mon, I think you can find a good enough bike to rent when you get there.