I don't know much about the background here, and I have no idea about your knowledge of cats, so if I'm telling you things you already know, I'm sorry.
First of all, URIs are not the only medical problem that causes cats to pee outside the box. With a thyroid problem, I assume he gets full blood work-ups done regularly and all other medical reasons have been eliminated?
Next, unless the smell is COMPLETELY removed from the place or item, a cat that pees outside the box will often continue to pee in the same place. Clothing, carpets, etc. should be treated with an enzyme cleaner before being otherwise washed. On the carpet, the enzyme cleaner (such as Nature's Miracle or Simple Solution) must soak completely down through the padding - all the way through to the floorboards or the smell remains. Anything cleaned with ammonia will continue to smell like pee to a cat. If you would like to visibly check for cat pee remaining in carpets, etc., just buy a black light. You'll be able to see anything on the carpet (even though you've cleaned it!) using this.
Does your cat have any mobility problems? Sometimes not being able to comfortably get into the litterbox is the issue. Does he have arthritis? Does he have to go far to get to a litterbox? Litterbox placement can be an issue. Do you have more than one litterbox? Does he have to go up or down stairs to get to it or one of them?
Some cats can be very picky about the litter. Ferals, for instance, often need to be induced to use a litterbox by covering the litter with dirt. Have you tried different types of litter already? Have you tried clay instead of clumping, one brand vs. another, shredded newspaper instead of litter, etc.?
Have you read through this VERY helpful thread? Inappropriate Peeing Problems Answered
At 13, it could be he has developed Alzheimers. Becky's Domino just posted her very sad experience with this horrible disease. In older cats, one of the symptoms is frequent urination outside the box.
95% of the time, a cat's peeing problem does, in fact, turn out to be medical. The first thing I would do is see another vet to get a second opinion. Once you are absolutely certain your cat is healthy (other than the thyroid problem being treated), and you've tried using multiple litter pans in various places, trying differnt types of litter pans, different litter pan placements, and different types of litter, and you're still having a problem, then perhaps consider containing the cat to one room, not in his crate.
You can try retraining your kitty to use the litterbox by containing him in a small room, such as the bathroom, or a large crate - one that is large enough to have a litterpan on one side, some room for him to move around in, and his food and water on the other side of the cage. Once he is using the litterbox, if he does so, then slowly allow him into the rest of the house, using multiple litter pans.
Containing him to his carrier is just out and out cruel.
BTW - when you say you feed him once a day, do you mean wet food, and he's allowed to free feed dry food? Cat's digestive systems are set up to eat many small meals a day. Unless this is on the advice of a vet and has to do with his thyroid problem, I'd consider feeding him more frequently, at least twice a day.