Here's the summary:
I started having some problems and read through a fair number of threads on this forum to see who has run into similar problems. I eventually found that my symptoms exactly matched the symptoms of having a bad contactor. That is, the thermostat would turn the system on and off as it should, but the outside compressor unit would never engage. When I took the compressors panel off to start poking around (and looking for bad capacitors), I found that if I pressed the contactor, it would start right up! Problem solved, right?
The wrinkle is: I replaced the contactor, and still have the same problem.
What should I troubleshoot next?
Some more ramblings:
To get around the contactor not working, I just ran the wire around it and bypassed it. So that the unit doesn't run all the time, I just use the circuit breaker inside to turn it off and on manually. Not ideal, but it is able to cool the house down when needed. One concern I have is that I may be bypassing some sort of safety mechanism. Perhaps the unit is designed not to run if the coolant level is low, or something, and I'm forcing it to run?
The unit is about 7 years old. It's nothing fancy and was replaced through a home warranty, which only had to be better than the (20+ year old) unit they were pulling out, so you know it's quality (

). It looks like it might be a Payne PH10 (PH10JA036-J)
The capacitor didn't look bad. I work on computers for years, so I've seen bad ones, and this one looked fine. But, I didn't measure it with a volt meter or anything. Should I?
A few possible problems I can see:
The thermostat isn't controlling things like it should, even though it seems like it is. (Possible. It's a Hunter 44760. Maybe 6 years old, nothing fancy)
The contactor I put in was bad, too (bought from a service department of a local HVAC service company)
I wired the new contactor in wrong somehow (but I only swapped one wire at a time...)
Some other control element is bad, somewhere along the way?
Thanks for any thoughts, and thanks already for the troubleshooting tips I already read!