I would definitely ask the owner before putting anything on the saddle or before having a saddle maker open it up to look at the tree.
You would not put product on a saddle in good condition that you had on trial, except maybe secretly to cover up some scuffs you accidentally made 
Put the saddle on the horse as is to see if the tree fits. If no, return.
If the tree fits, you have the question of whether you like riding in it. That’s trickier because it may take several rides at speed and jumping to decide if it really suits. And you can’t do that without some restoration.
I would suggest giving the billets a good tug
then sitting on the horse at a halt and maybe brief walk to see if the basic geometry of the saddle works for you. If no, return.
If it does work for you, at this point involve and pay a saddle fitter to check the billets and give the thing a good look over. And to double check the fit in your horse. The saddle fitter will be able to tell you if the foam panels are compromised or if the tree seems broken, etc
At that point if it looks like a keeper, get owner’s permission and go all out on reviving it. I’d suggest a couple of washes with something to inhibit the mold, white vinegar or Lysol solution and a few days in the sun. Make sure to get all the crevices and underneath too.
I wouldn’t soak good condition high end modern tack in neatsfoot but it is excellent for dried out or basically unfinished leather. Make sure to get the underside of the flaps too. Perhaps a couple treatments with neatsfoot especially on the billets, and then finish with a waxier ledersbalm type product on the seat and flaps?
It’s true you will spend some cash on the saddle fitter and time on reviving the saddle but then nothing is free 