The core is foam, so not prone to rot
. Don't know if long term saturation will affect the bond between the core and glass, however. Would suspect it would have no effect but nothing to back that up.
Trying to find the fasteners that are leaking will be a nightmare. The actual source of the water could be anywhere on the deck and the water just exiting at a convenient place. Probably best to start where the water is coming through and pull the plugs and fasteners in an ever expanding circle from around that position till the leaks stop. If you are lucky, it's the fasteners where the water is coming through that are the cause of the leak. I wouldn't hold my breath betting on that, however. Seal the fasteners with Polysuflide (Life Caulk, 101) or Polyurethane
(SikaFlex, 5200, 4200) caulk as these actually require water to cure. You will have a hell of a time caulking without making a mess. These caulks are sandable so you can remove it from the teak. The bung holes will be the problem, however. Possibly using Brown/Mahogany colored lifecaulk, would be the least obvious and use it to seal the bungs as well. Might want to experiment
on a scrap piece of wood to see what it looks like. The grooves in the teak deck are probably filled with Polysulphide caulk which is also called Thiokol, it's original industrial name.
Butyl would be a great sealant but won't stick to wet surfaces. So that, resin, and most other sealants won't work because of the water. Using resin to glue the bungs in is really a source of future pain. The resin will get into the heads of the fasteners and make it very very difficult to get a screwdriver to bite when you need to remove the fasteners.