The Grande definitely needs the airbox w/filter screen. Before I read your second post, I assumed that the mixture was too rich. Remember, the other vespa mopeds use essentially the same engine but with a slightly smaller carb and jet (12.10 with a 43). The Grande should have a 49 in it. Over time, I've experienced the exact symptoms (sometimes to greater degrees than other) with Vespas I've worked on as well as my 2 Grandes, two Ciaos and a Bravo...
It was caused by different things at different times. Here is what I have found to be the culprit (in no particular order):
1. clogged main jet
2. bad carb float
3. bad needle valve seal (caused once by a bad needle, once by a needle valve seat that had been widened by the previous owner)
4. bad HT coil
5. demagnetized flywheel
6. clogged idle jet (hard to find in a 12.12--it is a little hole in front of the brass tube in the venturi)
7. fouled/worn out spark plug
8. clogged fuel valve (petcock)
9. empty gas tank
10. air leak in carb (not installed fully)
11. ??? (I am working on a Grande for someone else right now and I can't seem to find the problem yet).
I think that covers it.