Wood – how to repair bi-fold door with fallen slats

To be honest, I'd probably be lazy here, and stick it back on with a thin film of a good commercial adhesive, thinly buttered on the back of the tile. Leave the old adhesive as it is. The old tile will fit perfectly. This is NOT something for superglue, but perhaps for something like liquid nails in a tube. Buy it in your local home center or a decent hardware store.

Use some tape (the handyman's secret weapon is duct tape, so says Red Green) to hold it in place, but any decent adhesive will have enough tack to keep a tile there while it dries.

Really the hardest part to do is to re-grout around the replaced tile. I'd not be at all surprised to find that available in a squeeze tube too. So I checked, and indeed it is, here, for example.

I agree, it sounds like a veneer that has aged over the years, dried up and is delaminating. One of the problems I see trying to repair the damaged areas with a filler, is that other areas may soon suffer the same problem. With that said, if you want to attempt a fix, you could remove all the loose material, sand off any remaining glue etc , then fill the voids. You could use a plastic wood filler, a thinset type of material or even a setting type drywall compound. Fix it muck like fixing a hole in drywall. spread the filler on with a wide knife, and sand it smooth after it dries. Be sure to prime and paint the entire surface before allowing it to come in contact with moisture or the outdoor elements. This may be a semi long term fix, but think you will continue to see problems in the future. Consider a new door soon.