Their inefficiency makes possible more nuance, because the lightly touched embellishment notes come out more faintly than on a modern accordion, where they're too crudely audible. The earliest model we have for our accordion style is Max Yankowitz, whose 1913 recordings gave us our foundation. There were other accordionists, too, and we've learned and gone beyond those examples. Like the early clavecin players before Bach, we basically use three fingers, though we include the thumb. The 4th finger is usually reserved as a krekhts finger, often touching the note above the main melody note to get that weeping thing, and the 5th finger is mostly used in emergencies'.
Ornamentation seems to be extremely important to klezmer music.
If there isn't, there should be. The entire gamut of the improvisational and ornamental gestures of the Baroque exists in klezmer music, sometimes called dreydlekh and shleyfer. Like early and pre-Baroque instrumental music, klezmer music derives its melodic ornamental style from vocal music, the difference being that the khazones and paraliturgical vocal style is the basis upon which klezmer music's gestures are built, and this includes a list of vocalisms which are verbatim replicas of weeping and sighing, called in Yiddish, krekhtsn. The early Baroque ornamental figure which is analogous to the basic krekhts is the Nachschlag, the difference being that the Nachschlag tone is heard in Baroque music, whereas in klezmer music it is stopped, or swallowed. This sounds similar to the sound made when you're weeping and your breath gets pushed out in thrusts.