You know I'm very much in Lhamo's camp and think a condo was more fitting for your situation, but since I've had a bunch of experience with them, I'll add that the only point of home warranty's are to replace REALLY broken things. There is no point to have them do "maintenance" on your furnace, i.e. wear a blue jumpsuit, hem and haw and say it's fine. And you're out some money for the copay. To really make use of it, you have to decide, and read between the lines, that the furance is REALLY broken. Like, somehow a sledgehammer fell inside of it and whacked a hole in the heat exchanger. Then you may convince the home warranty mechanic to put in a new furnace. Then again, they may accuse you of damage. Then again, you may now be sitting with a busted furnace for months while they order a new one. Or, you may want to look for a used broken furnace, put that in yourself, then have them inspect the broken one, and then come and put in a new one, knowing you can put the perfectly fine one back in if they notice your fraud. Then you can have the joy of throwing out two old furnaces. Do note though that for any installations like that, they do charge you directly undiscounted for any things "not to code" which they will invariably find. Also, you will want a modern high efficiency furnace and they will charge you through the nose to "upgrade" to what really costs just a couple hundred more.
Yup, home warranties are a joke. Your furnace honestly, sounds just fine if it is now working. Probably a good time to focus on other projects.