Yes, it might have tiny pilot jets that clog easily. Our SV650 was a hard-starter until I changed the jets from 15s to 17.5s. Beyond being hard to start, sometimes one or both of the old pilots would get clogged and it would run like crap.
I don't think it's your vacuum petcock. That would cause a delay in starting if you hadn't run it in a while, but the inconsistent idle points to a carb issue from my experience. I'm not familiar with that bike, but I'm guessing it has tiny pilots like ours did. Sometimes running an additive can help get things cleaned out and running better, but if anything is completely clogged, it would have to be taken apart and cleaned out.
Maybe you have already, but it might help to do some searching on your symptoms and that specific model. You might pull something up, and even find some steps to help correct it.
'96 DR650, '01 SV650, '10 Aprilia Tuono 1000R