I'm an electronics guy and I am always working on things I have never seen before. A lot of them are fixable, some are not. But I know my limitations, and I know when to say "you need to send this back to the manufacturer".
Any decent luthier should be capable of doing normal guitar electronics, including complete wiring of pickups, switches, and controls.
Now, the Variax is a different beast, but the local "authorised service center" is simply going to do board level replacements as well, not component level replacements. The one advantage I see with an "authorised service center" is that the work will probably be guaranteed. At a board level, the Variax is no more complicated than a standard guitar. The parts are hard to get, but not hard to solder, and no one, not even Line 6 head-office, is replacing components on boards. Not worth the time and effort. Just bin it, and put in the new board.
In this particular instance, the OP just has a faulty pot. Dead simple to replace. The problem is that Variax pots are non-standard. But, the part is available, and also dead simple to replace. Sure, in an ideal world these would go back to a local authorised service center, at not too high a cost, but for some people, that is simply not practical and as I say, if you have a decent local luthier, they really can't make the situation worse, and often will make it better.
It takes 3.5 - 4 hours the first time. It'll go constant red when charging. Pressing the battery test should have shown you something after 3 hours though. You might have a bad battery. Leave it charging overnight and if still nothing the next day, send the battery back.