Get your multimeter out and start probing voltages when it starts acting up.
...It also doesn't make sense since with the a/c on, the fans should come on as well regardless of engine temp. And why would they work on start up only to not work seconds later even with a/c on or engine over 220F...?...
Here’s something to consider (assuming the fans work like on 2nd gen. LHs):
The fans operate on two speeds. There are two relays that control that. Fans off is with both relays de-energized. Low speed is with one of the relays energized. High speed is with both relays energized.
If the low speed relay is not working for whatever reason (bad relay, bad wire, bad PCM), the fans will stay off when they should be running at low speed, and when high speed is called for, they will operate at a lower speed (probably very close the the normal low speed).
So - if normally, high speed is commanded at startup, the fans will run at something similar to low speed instead, and then they’ll turn off. Then, as the engine heats up, the fans will not turn on at all until the engine gets hot enough to call for high fan speed, at which time they will run at the pseudo-low speed, and will never run at true high speed no matter how hot it gets.
So, it’s a long shot, but, again, consider that as a possible explanation.
If the normal startup run speed is low speed, then you can dismiss my suggestion.