You can hire boats in lots of countries, but like you say, it's similar to bikes, when you're throwing yourself down a rapid that might just kill you if you get it wrong, you want to be in your own kayak, that's just the right size, you are 100% familiar with and trust, and is outfitted properly for you.
If you were going to Africa to ride down a big-ass mountain, where you could die if you got it wrong, would you hire some shonky African bike from the 1970's that you hadn't even seen, with dodgy brakes, no suspension and which was probably going to be the wrong size, or would you take your own?
Well it's the same with kayaking, only more so.