I hear you on the eyes, mine are shot close up, most of the time I'm soldering blind.

Something about the pca9685 that makes it easy, I've done other IC's with similar pitch but they seem harder.
I have one of these cheap hot air guns, I'm amazed at how long it has lasted and it's been used a lot with times running hours. I highly recommend, no need to spend big bucks. With that said it's still better to remove parts using that technique I showed, that's the best tip I ever learned. Never remove solder when removing anything, instead add solder.
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The pca9685 is quite rugged, I have very little trouble with them, I don't think I've ever seen a pin stuck on. It's sounding like you have a bad mosfet instead. I'm guessing you have similar being driven by the pca9685? Those can get stuck closed, when that happens the gate with the PWM signal will have voltage which can looks like it's coming from pca9685 but it's not. If you remove the mosfet the pin will react as expected. I would look at that first.
That adapter is nicely thought out, I guess at this point it wouldn't hurt to get some. I currently have a customer who's trying to adapt the light, he determined the signal inside the light is 0-3.3v, are you driving the adapter with the same or 0-5v? I'm hoping 0-5v PWM from your pca9685.