I would get a second opinion from another vet. It isn't necessarily a UTI, but that's a common cause and ought to be ruled out before you get too far into other conclusions. Your cat should also probably have a kidney function test, since change in litterbox habits can be a sign of kidney failure in older pets.
There's no guarantee it's a urinary tract or kidney problem, but unless the vet actually tests for that, you can't be sure it isn't - so that would be my first step. Just like humans go to another doctor for a second opinion, cats can too.
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