Home warranty companies usually have one, maybe two contractors who do all of their work. The contract(s) they use are the ones that cost them the least, so you can expect to have ****ty work done that is little more than a quick fix for whatever problem you are experiencing, especially when it comes to HVAC. Home warranty companies will fight tooth and nail to keep from ever replacing an entire AC system.
As for experience with American Home Shield, I had an AHS warranty when I first bought my current home (before I knew much about doing my own plumbing work). I ended up with a break in the primary drain line from my AC, and AHS's contractor literally cut the top and bottom off a 20 oz Coke bottle, wrapped the plastic bottle around the broken PVC joint, and then wrapped that with duct tape. I fought them for weeks to get someone else to come out and fix it, but they wouldn't because they only had one approved contractor in all of the Houston area - the one that had already been to my house. I ended up learning how to repair the line, did it myself, and then got them to refund my service call fee.