You used the "classical music" tag and within that style I think you will find an overwhelming tendency to include the third in the keyboard part even when the third is supplied by another instrument.
A rule of thumb I learned a while back is to reserve the use of open fifths or octaves (open meaning no third present) only at cadence points. That is specifically in regard to two-part writing, but I think it applies generally in other settings.
The sound of an open fifth will be found in other places - metrically weak, "horn" figures, musettes, etc. - but the basic, default sound is thirds and sixths.
The opening of this Haydn trio (Hob.XV:25) provides a good illustration:
Open fifths are found in metrically weak or cadence points while third abound elsewhere...
In the second half after the barline we can also see what happens when the third of the chord is in another instrument...
...notice that the open fifths in the red boxes are metrically weak. But the more instructive point is the
A dominant seventh chord. The third and fifth of the chord are played in the strings and doubled by the piano (blue and green) and then the left hand chord of the piano part includes those same tones again (orange.)
Clearly the approach is not to have the piano supply only the remaining tones of the chord!
... At first, I treated every part (piano L, piano R, flute, cello) as independent parts of a little "orchestra"
A trio is chamber music. In more sophisticated chamber music writing the parts are treated very equally, sometimes it's described as a conversation between the parts. The piano part should work pretty well on it's own, just as the two string parts should be strong melodically.
If the piano part doubles another part, it doesn't necessarily degrade the independence of the piano overall. It could be the doubling for instrumental color or emphasis, or simply because a particular passage doesn't require additional harmony tones.
If the piano steps back into a basic accompaniment role, it should still show good harmonic writing. Open fifths would not be the usual thing in such a case unless it is done for effect (horn fifths, musette, etc.)
Maybe it would be good to look at Baroque string trios where the keyboard part is written as a figured bass line. That's similar to your idea of treating the keyboard part as a single voice completing three part harmony. The right hand then filled in the chord or adds figuration and will certainly exhibit the typical harmony of thirds and sixths. If you look for examples with figured bass realization, it will show the bass line separately along with an example of how to accompany with the right hand.