In recent years there has been a great revival in the performance of early music and many performers strive for a degree of authenticity in their playing because we know more about how it was done.
Playing editions have changed in the last 50 years as more research has been carried out and we understand far more now about correct styles of playing and singing, playing at a suitable pitch, etc
What we should aim for maybe is an informed performance based on the playing techniques of the time and the capabilities of the instruments then to understand them fully.
Pianists are not banned from playing the music of the 17th cent because they do not have harpsichords of the correct design and era, nor are they banned from playing the music of Mozart because the modern piano is not like the lighter pianos of his time but they will gain so much if they are able to do so. Yes, it matters what our instruments sound like, of course . A piano will never replace an instrument of the harpsichord family and most pianists don't have access to harpsichords but we should try to play with sensitivity and in a style that would be recognised be the original composers and players I think.
Would you enjoy for example a party of modern day minstrels playing music of the Tudor age with a heavy vibrato on plastic recorders made to a baroque pattern, accompanied by a modern guitar pretending to be a lute, with the players dressed up like Robin Hood’s merry men straight out of a 1940’s film.
Music moves on and it would be difficult if not impossible to recreate even the symphony orchestra of for example Sir Edward Elgar’s time. If we travelled back to his time we would see wooden flutes with fewer keys, violins, violas and cellos strung at a lower tension, etc. The design of the instruments would be different and so the sound would be too.
When it is possible to do so we should listen to and try to understand why music sounded as it did rather than how we think that it should be played on an instrument for which it was never intended. Don't think of harpsichords as the second rate keyboard instruments that were used before someone had the sense to redesign them into pianos because they are totally different beasts.