I've been reading some musician biographies lately. By the time a biography is warranted, the musicians are usually in their later years, which means that their formative years were quite some time ago. In the case of the first crop of great jazz players, they came up in a time before the internet, before television and in some cases before radio and recording. Early rockers were not as limited, but recordings were sometimes scarce, and live TV appearances not that common.
One theme that runs through all these biographies is early influences. Interestingly, for some of the old guard there usually weren't many. Often as young enthusiasts they would make do with sitting outside a club where their hero was playing.
Fast forward a half a century and we have the complete opposite. You can now view all of the great players (with a few exceptions) any time, 24/7. Add to that a veritable avalanche of online lessons and demos and you have an insane amount of influence and inspiration to choose from.
You'd think that having few opportunities to hear/see good players would be rather limiting. And yet all the great jazz players seem to have fared just fine having absorbed inspiration from a relatively small number of players.
I recently tried to summarize my own influences, and it wasn't an easy task. I started with a very long list of drummers who I really dig. Yes, I absolutely love listening to these players. But when it came down to naming the people who'd influenced me the most, the list was extremely short, perhaps 3 or 4 names only. And that makes sense. It simply is not possible to emulate a long list of players. I also find that the influences are situational: if playing rock, I 'borrow' from my favourite rock drummers. Playing jazz, same thing.
The key here is to go ahead and be influenced. Don’t fall into the trap of avoiding influence -- that's a great way to stagnate. And don't feel you need to master a plethora of styles. Just be sure to pick up enough different material that, when combined, it expresses who you are and what you want to say musically.