A couple questions:
1) Is the motorized bicycle choice for fun, or out of necessity (cannot drive for whatever reason)?
Partly fun, partly budget - that $100/month I mentioned also has to cover all my transit expenses, which usually means walking or public transit. I've had a bicycle before, which I enjoyed; I'm quite likely to end up with a bicycle again, for puttering around my city and county. If I can add a motor to said bicycle which extends the range I can travel, reduce the effort I exert, and is reasonably priced, then it'll be well worth the cost.
2) What speeds are you looking to travel?
From what I've been able to read so far, 20 mph would be quite satisfactory.
3) How much are you willing to pedal if a breakdown occurs?
Lots.
I'm already willing to take multi-mile hikes at the drop of a hat; and once I get my bicycle legs back, will enjoy doing the same thereon.
4) Local laws? I thought I read somewhere that NY is not motorized bicycle friendly, but that might be within NYC.
I'm actually in southern Ontario instead of NY, where motorized bicycles are sold openly. If I can't cross the border, there's plenty to do on this side of the Great Lakes.
$2k is a very healthy budget.
It's probably going to be somewhat less than that - that $100/month is for all my non-essentials. I should be able raid my piggy-bank to manage $500 without too much difficulty by this summer; and it seems like it might be a good idea to get a few months of practice on a motorized bicycle locally, before taking a longer trip elsewhere.
Unless you are wanting to go the motorized bicycle route just for the fun of it, I would just get a scooter or used motorcycle and not have the worries of saddle comfort or reliability.
But if you are looking for a motorized bicycle and would entertain a non-standard bike, how about a tadpole recumbent? That would likely be my choice for a long distance motorized bike. A tadpole with a 4 stroke Honda or clone should be rock solid.
I'm not particularly familiar with recumbents, but am not constitutionally opposed to the idea. Would such a beastie have room for a good-sized backpack, panniers, or some other way to carry a similar amount of camping gear?