Hey Drew, This is a long post but I'm hoping it is thoroughly informative for you!
I actually have a property right now that is under American Home Shield (AHS). I'm currently out of state so I find it very convenient for me to be able to put the service request in online immediately after my tenant reports an issue, and they arrange for a reputable local company that they have a contract with to come out and assess the issue. Following that if any repairs are covered under the warranty, it is covered in it's entirety.
For example: My fridge breaks and my tenant calls me up and lets me know. I place a service request online, AHS emails me details and the company that is going to send someone out. That company will then contact you directly to set up an appointment.
One thing to note, as an "absentee" owner such as myself, I call up AHS after the appointment has been made but before the servicing takes place, and I pay upfront a $100 "service fee" with credit card. That way the tenant doesn't have to deal with it, just has to be at the house when they arrive. The person taking care of the issue will call you up and discuss any further repairs. This service fee is due for every service call you set up.
Pros: Relatively easy to use online service request, pay upfront and tenant gets it taken care of. If you have relatively expensive appliances that would be very detrimental to cover the cost of yourself if it were to break, this is a good plan for you. For a mere $100 the problem can be fixed in it's entirety, including replacing an entire appliance! They cover major issues with plumbing and electrical too.
Cons: Cost to cover major things in your household is about $635 per year. I would say if your home are relatively expensive, it is worth it to go with AHS because if an expensive fridge (or any other item) breaks it's replaced in that $100 service fee and you end up saving a ton of money. My unit is not that upscale so in 2-3 years if only 1-2 problems occurs then it really doesn't pay for itself. Right now I use it for the convenience and peace of mind of having that service available while I'm not near by.
My suggestion, and something I have been seriously thinking about now is perhaps start with AHS and save money as you go for future incidences that require fixing. The beauty of AHS is they tell you exactly what company they're going to use to deal with whatever problem you have. Collect the names/numbers of the companies they send out and keep it for your own future reference. Eventually you probably will have enough money saved that you wont need AHS and can manage to just call up these companies yourself and take care of it.
One issue I find a little frustrating is they'll occasionally call me during a service request and tell me certain things are not covered, and they'll ask for an additional amount to repair the entire thing (though it isn't usually some ridiculously high amount, maybe another $100-$200 depending on the problem). This doesn't happen all the time but it can be annoying. Be careful to read through the plan policies and see what is covered and what isn't so that you know exactly what you're getting for your money.
As a side note, the appliances covered, unless damage is the fault of the tenant, it is usually covered by the landlord. I'm surprised your co-worker has them pay for the service calls.
I hope this is helpful, please feel free to reach out with additional questions.
