Hey Mister!!
 I realize this is an oldish thread, but interests me so here goes.

 Did you ever get on the road?  Do you have a pic of the finished product?  How did it work out in comparison to the original?  What sprockets did you use - factory or different?
tom

Maybe here:

https://goparts.eu/en/genuine-oem/sym/motorcycle/wolf-sb125ni-l8-eu/2018/rear-rem-rear-wheel

 They show the parts at least.   Cam Purvis at scooterseals.com may have access
to other Sym parts.
tom

 I do not ride at night for the most part, so don't know how bright or dim the factory headlight bulbs are in general.  But, there are multiple LED options available that use less current to generate even more light.  I think.  But, again, the LED must be constructed so that its output is at the center of the reflector.  Just stuffing a bazillion LEDs into a 'bulb' doesn't do much good, except to light up the clouds, cows, and everything along side the road, while maybe lighting what is ahead.  It must be focused properly or all is wasted.
 If you look at the bulb assembly, you should be able to discern that it is 'keyed' so that it fits into the reflector only one way.  Any LED to replace that should also be keyed to insure the light developed is going to be reflected properly.
 You do not want to overdraw the circuit as it could blow a fuse or melt the insulation, and you would need room inside the headlight for the LED socket if you went that way.  Perhaps additional lighting, external to the factory, would be a better solution.  Look for auxiliary driving lights or headlights.  There are many on the market, and there are models that can be mounted to the handlebars.
 A CSC RX3 rider on the CSC group on facebook living in TX recently posted about his added lights that he was very pleased with.
tom

Did you go here:

https://www.shop.pbisprockets.com/product.sc?productId=551

 I did not fill it out, but for the indicated cost, seems very attractive.
You do need to provide specifics on center hole, bolt hole position & diameter, thickness and number of teeth(and size...)
tom

Luna, a trick that may help to remove the old grips is to use compressed air.  Get a air nozzle that spits compressed air in a small area, generally used to blow off parts, or clean shavings, etc.  Direct it towards the close end of the grip, so the air can get inside the grip.  It will tend to 'blow up' or expand the grip, or at least break it loose from the handlebar.  With the air flowing, direct it around the sides, top and bottom to insure the grip has at least moved, then attempt to slide the grip off the handlebar.  You may find a youtube example of doing this task.
tom

 Did you remove the idle needle adjusting screw?  It can have some 'stuff' at its needle end, the pointy part that pokes into the venturi, that can block flow.  The port to the venturi could be partially plugged.
 Did you pull the pilot jet out completly?  It has holes drilled across the 'tube' portion, forming an emulsion tube where the liquid fuel and incoming air get acquainted and start to 'foam', forming an 'emulsion'.  Bubbles of air get entrapped and carried on to the gizzards of the engine.  If the emulsion tube is plugged, the idle will not be very stable.  If the flow from the tube to the venturi is clogged or has a bit of blockage, again, poor idle.  Make sure you didn't soak the oiled air filter element.  Press between some paper towels to absorb any excess.  You want it wetted, but not soaked, and not 'drooling' any oil.  Smush it a few times, re-fold the paper towel, and repeat.  It doesn't need but a few drops distributed, and pressing will do some of that distributing.
 I found that a plugged pilot will make the engine jump up to a couple k rpms with just a close look at the throttle grip.  No need to even turn it. Makes for CRAZY riding when the engine zooms at a whiff of throttle motion, and then sometimes just takes off rpm-wise on its own.  Clean the pilot, the bowl, and the main jet, make sure the slide moves freely, and returns to closed/idle position without hesitation as you put it together.  If you didn't remove the idle mixture screw, you'll have to get a pick or a tool to dig out the plastic goo so you can fiddle, IF you think the mix is off once you are done.  If you are at high altitiude, it will likely be running a bit rich at idle, and perhaps on the main jet also.
tom

...
Just re-read your description.  I think you are not running on the pilot jet nor the idle circuit yet.  I tried to describe a wild idle that I had when the pilot was gunked.  It was adjusted, the throttle, to get it to run, but it was very unstable.  It was because the throttle plate was just at the low edge of the middle speed circuit, which was not able to keep the engine running at idle rpm, and jumped to high rpms all by itself.  That was a plugged pilot, and a mal-adjusted throttle idle setting in an attempt to compensate.  Get the pilot out, poke with a 'hair' from a wire brush.  They are stiff enough to poke through new 'gasoline' goo that has aged and set.  If you cannot see light through the pilot, it is still plugged.,  You should be able to run a brush 'hair' through from either end, with a little fiddling to get the wire to line up with the center of the orifice.
tom

Make one yourself.  If you have access to a hunk of wood, a drill, a Bernz-o-Matic torch, and some lead, you can cast your own end piece.  See on youtube, sixtyfiveford channel for explanatory example.
 You can purchase cable 'inners' of a length you like, or an emergency cable from 'zon or 'bay that will get the job done.  I have come to the conclusion that cables are not as exotic to make as they seem.  Watch a few tubes on them, and give it a whack.  What do you have to lose?  Time.  But life is all about time(among other things) and knowing your bike, how it's made, and especially how to fix it when 'things' happen alongside that road somewhere west of Laramie is what adds to the spice and zest that gets you up for that first cup in the AM.
 Make one.  You will be proud, and can relate the story to the rest of us schlubs...
tom