The Genoese were famous not only because they were skilled professionals, but also because they were widely available in relatively large numbers (if you had the cash). The high numbers of professional crossbowmen seem to have been a result of the city of Genoa's naval capacity. Galleys, under Geneose law, were legally required to have certain numbers of crossbowmen on board. Of course, the existence of such a law doesn't guarantee that it was always followed to the letter, but there's no doubting that the use of the crossbow was linked to the city's naval/mercantile success.

There weren't "copycats" because there wasn't anything particularly unique about Genoese crossbowmen, beyond the fact that they were professional and available for hire. Practically everyone utilized crossbowmen in medieval Europe (including the English), but not all of the available bowmen would be experienced professionals. Professional troops are expensive to train, equip, and maintain. If urban militia crossbowmen were not adequate for a combat role, why bother going to the trouble of developing your own capacity for professional crossbowmen when you could easily hire mercenaries? The Geneose, although popular, weren't the only options. The Teutonic Knights hired Genoese crossbowmen on occasion, but also recruited crossbowmen from various German cities.

The idea that crossbows are "easy" to use in comparison to longbows isn't very true. While they take less raw strength to operate, loading a medieval crossbow is definitely not an easy task, especially under the pressures of combat. Crossbows themselves were fairly expensive and their production was generally of great interest to the medieval state (whether that state was the Teutonic Order, an Italian city-state, or the Kingdom of England). The use of the crossbow seems to generally have been reserved for either professional troops or urban militias, formed from relatively well-off burghers and townsmen.

The Genoese were definitely not the only soldiers to carry pavises. There's actually a fair number of surviving examples of late medieval pavises scattered across American and European museums, many of which are highly decorated. Check out this one from the Metropolitan Museum of Art in NYC. Pavises weren't exclusive to crossbowmen, either. Check out these Hungarian noblemen from a 16th century print commissioned by Holy Roman Emperor Maximilian I. These German swordsmen are rocking the puffy sleeves and hand-pavises, as are these knights.