So yes, you can ship your own cat. At least in some circumstances.
I lived in Chicago, so IAG (BA's cargo arm) is based there and they were happy to take the animal directly from me without an agent on the US side.
I read all the instructions (carefully!) and managed all the paperwork and health check and certifications myself on the US end, with the help of my vet who is a USDA certified vet and understood the process. It isn't difficult, just quite detailed and you must absolutely get those details 100% right, and it's a little nervy on timelines, especially waiting for APHIS to get the certificate back to you in time for the flight. If you miss that deadline, then you have to start again, but that would be exactly the same with an agent. IAG put our cat on the same flight as us but we (IAG handles this) had to book a slot at the Heathrow Animal Reception Centre (HARC) about 8 weeks in advance (although you cannot book the animal's flight until 13 days out). There are limited flights they will put the animal on. For example, the one after ours was not ok the animal's flight until 13 days out). There are limited flights they will put the animal on. For example, the one after ours was not suitable since they were carrying dry ice as cargo and that would not have been safe for the pets. I delivered the cat to the cargo area at O'Hare (having scoped it out the week before so I knew where to go) and then we went to the passenger terminal. The IAG guy doing the loading kindly called me to confirm that he had personally put our cat on the plane, which was reassuring. The HARC were very responsive, and when I emailed them a while after landing they responded quickly that they had our cat, he was fine, and was in the waiting area with food and water while they did his paperwork. You do need an agent to clear customs and pay VAT on the UK side, but the cost was quite modest. I used JCSLivestock, the one mentioned by IAG. They were very responsive too. I also paid them to deliver the cat to our Air BnB, which was well worth doing since it takes them a few hours to clear customs. The cat's ticket (freight?) cost about 1,000 USD, but on the bright side that was about 3k less than paying an agent to do it on the US side. There were some modest costs for the vet check and health certificate, too, on the US side.
Anyway, all's well that ends well. It might be more difficult and worth using an agent if you can't easily get to the air cargo site, or you have multiple animals. Dogs also require things like worming. Having a vet who is USDA certified and knows the process at least somewhat is very helpful.