The company I worked for when I lived in Raleigh was Wag the Dog. I can't speak about the quality of all the sitters on their payroll, but the CEO/manager/owner is a very nice woman named Crystal who's probably the best boss I've had in my career.
And yeah, I can understand why having someone else look over your pet with medical issues is a scary prospect. I've been rather lucky in that regard. Even with pets that had previous medical problems, I've not had to deal with them while looking after them, for the most part. The one exception was last year, where an older dog had a seizure between the time the house sitter left in the morning and the time I arrived for the walk in the midday. I arrived to a medical emergency, and by the time I got him to the vet, it was too late. I know, logically, it's not my fault and there was nothing I could do, but it still sucked being the one to deliver that news to the client.