I also love having teak. But I removed it and put non-skid on the foredeck. For two reasons - firstly there were soft spots, indicating rot of the balsa core. Secondly, teak gets very hot in tropical sun. We replaced all of the balsa core in the foredeck, and about 80% of the balsa was wet or rotten.
However, my sides decks and cockpit remain teak. These areas are partly shaded, so the heat issue is much diminished. I have done some localised repair, and have no doubt there are some small areas of wet balsa. But no really bad areas so for the time being I am keeping them. Repair was limited to remove the teak, clean the underlying GRP and inject thickened epoxy into the screw holes. Then bond the teak back down, re-drill the holes and insert screws and bungs, and finally re-caulk the seams. Its only been a year but no problems yet. Eventually I will need to lift the teak and remove the balsa core, but I'm expecting this to be quite a few years down the track.
For the OP, I'd try to get a feel for whether there are soft spots and how extensive they are. Hammer test is OK for a start. If you are able to remove screws and the teak strips intact then I would do that for any suspect areas. Then use a drill to put in some holes to test the extent of wet core. Just closely examine what cuttings the drill bit brings out. It doesn't really matter how many holes you drill, they are easily filled with epoxy afterwards. We started in a soft spot and just worked out from that until there was dry balsa. You may be able to get a lot of years life by doing what I described above. It depends on how big an area has rotten or wet core. I would not rush this - once you have ripped all the teak up its gone for good!