Potential detractors of period instruments in Mozart, who might opine that a lived-in interpretation must belong to our own period and not some past era, ought to realise that their argument can, ironically, be used to contradict them: one now hears more and more performances of eighteenth-century music on "old" instruments, and fewer and fewer on "modern" ones. The result is that it is above all the "neo-Classical" performances which are truly of our time, while the "neo-Romantic" approach is beginning to belong to the past.
It is of course possible to do a performance of Bach or Handel or Haydn in a "neo-Romantic" style -- large orchestra, slow tempi, Romantic-style phrasing -- but it no longer seems to come naturally; doing a big, Romantic "Messiah" is a conscious choice that requires as much re-thinking and as many conscious choices as a small-scale, fleet "Messiah" used to do. Playing a Mozart "andante" at a relatively fast clip is the common practice today; playing "andante" very slowly, as Furtwangler and other Romantic conductors used to, is an act of historical reconstruction. Because the HIP style has become so prevalent, it no longer seems to be particularly "historical"; it's just the style of our era.
Of course, HIP never was entirely "historical," despite the claims of some HIP artists who liked to sell their records by claiming that they were providing "authentic" Bach or Handel or Beethoven. While HIP is based on historical research -- reconstruction of old instruments, more careful attention to what tempo markings meant in a particular time, using vibrato sparingly -- the end result is a late 20th century style of performance that would probably sound pretty foreign to Bach or Beethoven if they came back today; the clipped phrasing may have some historical basis but it also sounds (as some have commented) influenced by the neo-Classical music of Stravinsky, and other attempts at creating modern music out of older stylistic elements. And of course, the best interpetations are dictated by the artist's own ideas about the music, not the ideas he's picked up from some 18th-century treatise; the result is that the best HIP artists will often do things that would not have been done in the 18th century but which sound right for the music. The most boring HIP performances and recordings are the ones that do counter-intuitive, anti-musical things for the sake of spurious "authenticity," like those Beethoven symphony recordings that never modify the tempo at all on the mistaken belief that respecting Beethoven's metronome markings means never departing from them for an instant; but these aren't the performances that are likely to get reissued 25 years down the road.
Ultimately, the reason HIP is a "modern" style is that it was created not to make music sound old, but to make it sound new. The granddaddy of HIP, Nicolaus Harnoncourt, was a cellist in the Vienna Symphony Orchestra for many years. In the '50s, he started to wonder why Vivaldi was considered such an important composer in his time, when his music always sounded so boring when the Symphony played him and was given to music students as an example of "easy" music. So he started doing research into the way music was played in Vivaldi's time, and essentially realized that if you used the instruments of the period, and certain playing and phrasing techniques that had been lost, the music wouldn't sound boring; it would sound fresh, inventive, and interesting, the way it must have sounded to 18th-century audiences. So he started his own orchestra, the Concentus Musicus Wien, devoted to playing old music on old instruments with old playing techniques -- but with all the "old" stuff used for the purpose of making 18th-century music sound fresh and new.