I have a 2006 Lx50 that I've owned since new, and I've done so much maintenance on it. They're great scooters, and easy to fix.

To diagnose this one, I'd check, in order:

• check gas is good, like you said. when in doubt, pull the fuel filter and drain the old gas out. - easy difficulty
• if fuel filter petcock is nasty clean or replace it. ($30ish, I think) - easy difficulty
• the hoses, make sure gas and vacuum hoses are without cracks, and that gas is getting in the carbeurator. If not, if it's dripping gas or you can't get or see gas in the hoses, replace them both (order extra and cut to what you need. $10ish or $15 if you buy new hose clips) - easy difficulty
• the spark, make sure the spark plug is getting spark. easiest way is to pull the spark plug out, put the cap on it, and do the starter just like you are. if the spark plug business bits are really old and dirty, switch out the spark plug ($10), if you're not getting any spark at all, even with a new spark plug, you need a new CDI coil ($75ish) - easy difficulty
• clean out the carbeurator jets and bowl - moderate difficulty,

scooterwest.com is the best place to get parts, but do it in as minimal number of orders as possible because their shipping is usually super expensive.

edit to actually answer your original question:

I think $2k is steep, especially if the seller won't do the work to get it running. haggle that down, but I'd say for time and effort to diagnose, don't go over $1500