In the folk traditions Im interested in, Britain and Ireland, Western Europe and Eastern USA, piano is mainly an accompaniment instrument when it does appear. A lot of that is bass/chord vamping. As Anyanka says, this can be too heavy and repetitive - but just like too much bass/chord on accordion basses!
On the other hand there are some accompanists whos playing is a delight, while generally playing in that format. Charlie Lennon in the Irish tradition comes to mind.
Charlie kicks off at about 38 seconds here
Both tune and accompaniment can be done on piano in the Irish tradition.
Heres Patsy Broderick
https://www.youtube.com/watch?v=VgiWQRj94wM
In the States George Winston has a style hes named rural folk piano - its a nice listen, but does it fit the label?
Significantly, hes self taught on piano.
Heres a lovely trad English tune on piano,
Folky in its way but also somewhat reminiscent of the pre-WWI English pastoral composers, who used English folk tunes of course. Full circle?
Im no pianist but I like playing traditional tunes on piano. I just tend to do a mixture of arpeggios and spread and block chords in the left hand, often using the keyboard three chord trick, ie in C major fingering CEG BDG CFA, plus minors.
But practicing CBA takes priority!
Id say piano can certainly be used in a folk context, but I dont think it makes it a folk instrument.
OTOH, there are plenty of videos on Youtube teaching chord piano and therefore players whove come to the instrument without notation, but via chord shapes, and by ear tunes. Is that a folk tradition in a digital age?