Any tricks to remove drivers from a (sticky) cabinet

Some manufacturers use a type of Goop/non-hardening sealant material to make the speaker secure and also prevent air/sound from going around the speaker.  In my past experience, the best I could do was using a small-wide screw driver and a firm putty knife, and went very slowly around the edge, prying only slightly each time I went around until I had pulled it far enough away from the sticky-stuff and to pull it out. This is not professional advice, just my experience. Hopefully it helps.

I've had similar problems and found that, with some speakers, it's easiest to remove the binding post plate in the back and push the drivers out from the rear.  Usually, this will allow me to remove the woofer(s) and the I can reach my arm up through that cut-out and push out the other drivers as well

If the removal of the binding post doesn’t work, and you’re forced to trying to pry the driver out, go to an auto parts store and buy a set of auto interior trim removal tools. They’re usually about $15 and come with several sizes and shapes of plastic pry tools that are designed to not scratch car interior trim, door panels, leather, vinyl, etc. I would go that route to prevent marring the leather. 

you can use a small inside caliper, into screw hole, expand tips and pull from the screw holes, not the edge.

$12. be there tomorrow

or improvise to pull from the holes, not pry the edge

using alcohol, you need to position the speakers to control gravity, i.e. fluid flow. you can lay the cabinets on their back, so fluid won’t run down the face, get an artist’s little paint brush or ... and try a spritz of alcohol, start swishing a bit inside the screw holes, wait a while, then tilt to use gravity to control the flow of fluid, some from the outer edge in any small crevice, bit by bit...