Should I remove the weather stripping type material and put new stripping on it? Maybe squishy thermoplastic?
***Additionally, this will be like the second or third time I've removed the wood panels. What are some good screw fillers you've used that will make them tighter and potentially repair stripped holes if that occurs?
I agree with Frank that the access panels should be attached with T-nuts or Hurricane nuts.
Since you have stripped them out, and wood screws are not keeping the panels air tight, they are a better solution.
I have not found gaskets needed when using T-nuts.
If your construction is such that it does need gaskets, you could use Mortite putty, it will be air tight and does not permanently set. Window glazing rolled thin like an earth worm works too.
Stripped holes can be repaired by mixing fine wood router dust with wood glue, and injecting the mixture with a large bore syringe, or simply apply glue on as many toothpicks needed to fill the hole and tap them in with a hammer.
If you have other leaking parts of the cabinets, you can pour in some paint and tip the cabinets around until it seeps in to the gaps. You may be able to find "oops" paints (mixed to some odd color) at some hardware stores.