> .... the safe sat in a barn for years ....
Exposed to urine/ammonia fumes? This scenario just might be desirable and may have lent itself to giving the wood its dark coloration.
A note about the possibility of having an underlying clearcoat: It doesn't have to be a dedicated clearcoat, like shellac, lacquer, linseed oil or tung oil. It could be only wax applied to the original raw wood.
The pie safe doesn't look to be a high end or super-finely crafted piece. Generally, they were utilitarian pieces. Yours looks to be of pretty good craftsmanship, carries itself well and may have more value as a folkart piece, as compared to a finely crafted antique cabinet/hutch. I would assume any upper trim or "head" molding, as well as any base molding, is as uncomplicated as the half-round moldings of the screen panels.
You may get exactly what you are aiming for: 1) a good used look (rather than the term distressed), 2) a minimally invasive restoration, 3) increase (rather than decrease) the value, 4) (a bonus) end up with a pretty good showpiece, after all, if the paint can be removed without damaging that, what seems to be an, underlying coating or patina.
1) The type of wood may not matter, if there is some kind of underlying coating/wax under the paint.
2) In tackling the project, don't view the whole piece in totality. Approaching the project, that way, may contribute to your being overwhelmed by what-all may be eventually involved in accomplishing the whole mission. The whole job may consist of ~350 small jobs. Do the small jobs, one at a time.
2.a) Remove one piece (a molding) and work on it, only, i.e., clean it up, see if the paint can gently be scraped off (without damaging the (possible?) underlying coating/patina), etc. You will likely discover what needs to be done (and how difficult) with the remaining pieces. Work on the remainder of the pie safe in this fashion
3) Once it's all "cleaned up", I would, then, evaluate all the jointery, missing pieces, broken pieces, etc., to see what aspects need to be worked on and/or repaired, plan/plot how to *approach any needed repairs, etc. *Approach: If you have to disassemble something, to make a repair, will you be able to easily reassemble it, without having to do some other manipulating to some other unrelated (intact?) piece?
3.a) Push (a bit) on the top, side, of the safe, to see if the whole wobbles or shifts (major joints loose?). Do the same with a push from the front toward the back..... i.e., is it structurally sound.
4) Is everything level, does it sit on the floor well, will it need adjustable casters, are the feet (if any) stable, etc.?
Take your time, do a good job. I like projects as this.