Not sure if white glue would be adequate as I believe it requires clamping of two wood pieces for the best bond. Doubt they are really made for structural gap filling or penetration on their own.

Mind you, I definitely do not have experience with all the different "Wood Glue" products out there. One such is Gorilla Wood Glue, which is a PVA similar to their more popular product I use for hinge installation, (just like CA they require moisture from the air to activate). Also, unlike the white glue of old, once cured that same moisture or wet environment is not supposed to reverse the process, although I have seen some get soft if heavily exposed to water.

I strongly suspect a basic "White Glue" or a contact cement are used, at least in R/B and other ARF products I've had across the table. Not a rigid bond and VERY little penetration of the wood at all. Both are counter to our requirements. Your other problem would come from any existing glue, even if separated, as it would prevent new adhesive from access at the bond line, directly where you need it.

I have heard of a few different ideas, but personally would not try them unless I had some method of contouring the stabs. Mixing standard Gorilla Glue with water will over-activate and make it expand like the spray insulation you get at the hardware or building supply, (also another option that has been tried). It fills the entire internal volume AND bonds the parts together. Unfortunately that expansion will continue unchecked and inflate the stabs like balloons unless there is something to prevent separation.

I would at that point pretty much decide that cutting at least the lower surface away to check and correct any errors would be far less trouble and increase my confidence. When I did mine I opted for FLOX, (Flocked Cotton) mixed with epoxy to fill the gaps as it makes a structural bond.

As mentioned previously, I found Ultra Cote is a perfect match and all I have used for years for its superior contouring and covering characteristics so it was relatively easy to do even the entire wings. Elevators and stabs use exactly the same technique.