Fixing the root cause is easy, just replace the petcock and change the oil. However, I can tell you exactly why the mechanic wanted no part of this.
If the bike was running on thinned-out oil for a long time, there's no telling what damage was done to the engine internally. In the best case, no serious damage was done and you'll just have to rebuild the engine at 90,000 miles instead of 100,000 miles (to pull figures out of thin air). In the worst case, your pistons and cylinders are shot, crankshaft bearings have spun, and the camshaft lobes are scored. Your engine is probably somewhere in the middle but you won't know where without a complete tear-down and inspection of the engine.
The other thing about working on old bikes is that you dont know what else is going to break when you remove parts to get to the initial problem you want to fix. It can sometimes be a massive rabbit hole - especially when replacement parts are NLA or difficult to find.
I was lucky in that the shop I worked in was also a wrecker (boneyard, breakers, whatever) so we had access to a stockpile of parts bikes to obtain parts from. Even then, when you charge for labour at reasonable rates and parts at below cost people were still thinking they were getting ripped off.