Dennis,
I'm not professing to be an expert on this subject, just that in my years I have never seen hatches sealed with anything other than caulk/sealant in the perimeter gap. Could the rubber pieces you saw have been put there to sort of "shim" the hatch? Or maybe it was there as an attempt to keep the hatch from squeaking when being walked on. I don't know, though - maybe it was an attempt to seal the hatch that was a short-lived option? Meaning, it was an idea that they tried out to see if it would work or not? Again, not sure - just thinking outloud there. But, personally, I don't any reason to start messing with other ideas when the perimeter caulking works so well. I certainly don't see a piece of rubber being a good, long-term seal in this application - water would still find it's way past the rubber. Foam rubber would work better (as a seal) as it's compressible and would take up the irregularites between the floor and the hatch edge.. but it would lose it's compressibility and flatten out over time.
Inspection plates... filling the holes with epoxy (possibly thickened or some glass, depending on the damage) SHOULD work. But certainly using thru-bolts and nylocs will solve the issue. If you don't need to pull the big hatches, I would think you can install the bolts/nuts through the inspection hole? Another thought... rotate the plate and redrill for new screws. Still another thought... a good adhesive sealant (NOT 5200... way too strong) will actually do most of the work. A little on the screw thread and under the screw head will keep the screw in place.