Solid wood filler?

Auto body filler, which is epoxy with the consistency of putty. It sets up in minutes, you can sand it in 30 mins and it's preferable to do it then if needed, because it will sand easier then when fully cured.

Have you thought of drilling out the hole to a larger size and inserting a piece of metal, such as a pipe, where the L bolt fits tighter? That would hold up better than a wood filler.

There is epoxy wood filler - but expensive and might not work with your caulking failures .. and would be questionable for a heavy gate that requires a 3/4 inch lag .. perhaps yours is 3/8 or 1/2 ? If you drill-out the hole and glue in a "dowel" < eg. broom handle> - the dowel will turn free as you screw in the lag - you would need to square it . Moving the hinge or replacing it with a dufferent type might be the best bet.

John T.

If the wood is good - not rotted or soft - you could do this...

1. Chisel out the wood where the fasteners go making a hole that is 1/8 -

1. Get some epoxy resin, hardener and Cab-o-Sil filler. The latter is fumed silica, very light and fluffy. and should be available at any place that sells epoxy in cans (as opposed to tubes). You could use auto body filler but the resin is polyester resin, not epoxy and the filler is (mostly) talc. It should work but not as well as epoxy and not - maybe - for as long.
2. Pour out enough epoxy to fill the enlarged hole. Add the filler to it and mix it in. Continue adding filler until you have a paste about as thick as peanut butter...thick enough so it stands up and doesn't run.
3. Add the proper amount of hardener to the paste mixing it very well.
4. Pack the paste into the hole.
5. Insert the fastener into the paste. Once inserted, do not move it. Wipe off the paste that squeezes out.
6. Wait a day then sand if needed and paint.

If the wood is rotted, you could do the same thing, just make the hole big enough to get to good wood.

If you have a stainless or aluminum fabricator nearby you should be able to talk them out of a coffee cup or two of filings. Both will be pretty sturdy filler. Just be sure it is not ferric (magnetic).

WaterWeld putty from JB weld.

Would a gate hinge bolt that goes through the 2x4 be an option? Something like this:

Or pack steel wool into the hole.

Can you move the entire hinge up or down a half inch and just drill new holes?