Like automobiles, I suspect accordion manufacturers thought ease of repair and maintenance a secondary consideration when designing the things. Apparently, PVA was discovered/synthesized in 1912, but I don't think it was used as an adhesive much in those days. Pre-WWII furniture makers preferred animal hide glue, aircraft manufacturers liked casein glues and bookbinders used all sorts of stuff, including starch-based adhesives. These preferences were to do with the different properties of the adhesives.
Starch-based adhesives can be 'reversible' - although just how easily, and with what collateral damage, I don't (yet) know. Note that flour-and-water paste, often loosely called 'starch adhesive' is not considered reversible - the gluten in flour makes cross-links. Hide glues are reversible, with heat (often steam). PVA sets to what is in effect a thermoplastic substance, and can be softened with heat. (You can't really wash bellows, to get rid of the stuff...).
PVA is a wonderful adhesive, and will surely do the job of sticking bellows tape. But too well! Also, removal is a problem, because it relies on its adhesive excellence by soaking into the substrate's fibres. Re-application of PVA over cured PVA may disappoint, because the new layer won't adhere well to the cured stuff.
Since accordion manufacture seems to me to be more the product of artisan than scientist, I'd bet that tradition played a large part in adhesive selection, at least in the early days, and I suspect that something like a starch-based adhesive, or wallpaper paste, was used - not new-fangled PVA.