I think there's more to it than just how new or old is the piano - certainly more than how portable it is. I think its an issue of even temperament. There's no doubt in my mind that certain traditional melodies take on a different character when accompanied on a piano - one that springs to mind would be "Underneath her Apron" with its "ambiguous seventh" (I think I made that term up!)
Here are a few random and not very well thought out ideas:

Why is it that the "schoolroom" folk songs that the older among us will remember do not really feel "folky" - how often do you hear anyone sing "Early one Morning" ij a singaround? - or "Dashing away with thte smoothing iron"? - is it because of the formal effect given by the piano arrangements?

Why do the sparse three-part "a capello" harmonies of the YT, or Coope Boyes an Simpson, or many other similar groups sound so rich and full? - is it because being unaccompanied by even tempered instruments they settle back into true "pure" harmonies with all the attendant harmonics or overtones?

Does the same apply to the spontaneous harmonies in a singaround?

How many other instruments are tuned to even temperament - concertinas? - guitars - especially guitars; when you tune your guitar from string to string to intervals "heard" in your own head does it end up in even temperament? - is that why when it seems perfectly good on a chord of C it's rubbish on a chord of D?

Do string players, not constrained by frets, improvise in true or even temperament?

I feel the need to hear from John in Kansas who I would guess from earlier threads seems to know all about this kind of stuff.

SJ