Maybe something along these lines?
https://www.amazon.com/Mirrors-Extender-Riser-Adapter-Scooter/dp/B007S048FI
https://www.ebay.com/itm/1-Pair-Mirror-Extender-Adapter-With-8MM-Bolt-Universal-Fits-Scooters-Sportbikes-/264468776991

Or if you get a slipstreamer windscreen it comes with the  option to raise the mirrors a bit.
https://www.partsforscooters.com/172-83_Scoot_66_Windshield for instance. (check the hardware pic)

There are also alternative mirrors and such in a variety of arm lengths and shapes.

Cheers
Tromper

This is where I'm at my limit.  I do not have the book on this one.  This bike was not released in the U.S. & I do not have a tech manual for it.

RE: Valves This is likely the same as the couple similar 4 valve liquid cooled 125s from SYM I do have the book for, but it could easily be a bit different, so I can guess that the valves are the same spec, but that's a guess and I'm not the one who'd be out of luck if that's wrong so I ain't going there.  You'd need to look that up.

There are more than a few videos on setting valves.  Your joymax will be a bit more involved since it's going to be a four valve vs. the normal two valve.

The Vacuum system is going to be the seal for the carb to the intake, normally a rubber boot with a clamp on these, & the rubber lines running off the carb for the most part. There are a  couple others, but it comes to the same thing. check them one at a time, replacing them doesn't hurt just be careful so you don't lose where they go.

You'll probably also have a PAIR/EGR system which will be some vacuum lines off the carb as noted above & a hard tube that runs from a reed valve on the valve cover, over the valve cover, that goes to the exhaust.
Those usually aren't a problem, but if they crack, or clog they can cause running issues.

Far as the carb goes, hopefully this isn't quite as finicky as the HD200 carb is known to be, but there are some videos on that as well.
Normally just clean the jets and you're good.  You might even get by with a carb cleaning solution like seafoam, but it's tough to say without being hands on.

This stuff is bread and butter to a mechanic, it also doesn't usually need a lot of adjustment at on SYMs in my experience but it is stuff that needs to be checked. not sure what costs and such are in your area, but it might be worthwhile checking into what doing a run though on that would be locally.  If you do handle this yourself.  I applaud you and suggest you take it one piece at a time and take your time the first couple times through.

My apologies that I can't be more specific on this one.

Have ya checked the valves?
Checked for vacuum leaks?

Ditto that, & I even have 5k+ on a Dayco no issues so far.
I also have 3 bando OEM belts I replaced that are well within spec and I'd reuse without hesitation.
Definitely something odd going lie that and that badly.  It does look like you should be able to chip the debris out of the splines, but sounds like ya tried. 
I'd be tempted to try a little heat (don't want to ruin any tempering) or even cold to see if that'd cause the debris to loosen a bit, but that's a very much at your own risk kinda thing; albeit ya ain't got much to lose here.

If ya decide to part it out lemme know.

Either that was the wrong belt, or some cheap Chinese clone or not installed properly.
While I replace the belt at 5K miles there are plenty of people who run them 12K plus miles. Normally they start to stretch or slip before they just break.
That shop owes you a new scoot. See if you can find any kind of markings on it.

Dang that looks messy.

If that's metal in the splines like ya note then I'd be inclined to fix it by replacing it.
Unfortunately that's pretty involved.  Not something I've done personally, so I'd probably check to see what a shop would charge & if not too extravagant pay for it. I'd also ask 'em about other rebuild items while they have it open as well.

If that's too much $$ and I had the time I'd dig into it, or get different bike if I couldn't dig up a used motor (these aren't that common but you could get lucky).

How many miles on the belt when it blew?
Was that a Bando/OEM or ?

Well heck.  Annoying enough.  I suspect it wasn't on their agenda either if you're going to leave it there. 
Dealers (around here/heavily urbanized) don't like to keep extra bikes around since space is at a premium.

I'm guessing the tire size isn't real common in your area?

I took the Sym to the dealers yesterday, it was booked in because I could not remove the rear wheel. I arrived,as arranged three weeks ago, at nine a.m.  The dealer said the wrong tyre has been sent, so it could take a couple of days for the correct tyre to arrive, this, despite me having told him the tyre I wanted just four days ago. I now have to wait for them to contact me when my bike is ready.

https://www.manualslib.com/manual/804351/Sym-Hd-125.html maybe?

Check the kickstand switch.
It might be sticking.