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Hei guys.
It's been a while since I was on a forum so I want to share experience I had with my AG 50 after 54.000 kilometers.
Since I bought it new, had no serious problems so far except regular maintenance and it still goes very well.
Had also some crashes, broken right rear footpeg, windshield and top case bracket.
Scooter is standard with oem head, piston, carb, etc.
Went through 6 drive belts, bando in original size proved to be the best in everyday use so far.
6 or 7 sets of rollers + one new drive variator set(front one), rear driven pulleys are still original but needs new bearings and grease seal or complete new set. 
6 new sparkplugs, ngk iridiums works best for me.
new rear tyre, front one still original and good!!!
Replaced piston rings and head gasket.Burning a bit of oil but it's going good.
Regular oil changes at 2.000 km's, Motul scooter expert 10w 40
Regular transmission oil changes, motul scooter gear 90
Replaced front headlight bulb couple of times and some dashboard bulbs also.
New speedo cable.
Front brake pads replaced 2 times, rear one's original.
After 5 years, battery is still original one and serves well but scooter is in frequent everyday year-round use.

I'm very happy with scooter and people often ask me how is it possible that 50cc scooter covers that mileage.
I just do regular maintenance and it keeps going.

As you described, I think the problem is in the throttle cable.
Other cause could be cable connection to carburetor.
Third cause could be throttle handle on steering bar, but you said handle could move freely while throttle is wide open.

First, remove upper handlebar cover, four screws and one on the front.
When you remove it, remove two screws at the back of the right handle holder(black piece where is starter and light switch and it holds right rear view mirror).
It will come in two pieces and you can remove throttle handle. Disconnect cable and you'll see small rubber dust protection.
Try to put some oil in the cable.It should be lubricated every now and then.
Then, put some grease on the right of the handlebar where handle sits.

Put it back together. It's not so hard to do this 

if you have 12'' agility 50, than cut through cdi box. you'll find instructions on this forum or youtube,
if it's 16'' agility 50, than you should cut some wire. not sure which one but instructions could be found here as well.

good luck 

reg, as I know, belt shouldn't touch kick start half moon, bendix or cvt cover.
maybe belt pieces was flapping inside making marks everywere before it snapped. happened to me twice
took me half an hour of cleaning the pieces and sticky black dust

my advice:

-buy quality belt
-clean and lubricate torque driver for smooth operation regulary
-clean inside of cvt cover and variator system. you don't have to be thorough. compressed air works for me just fine
-when installing belt and tightening front variator nut, rotate variator a couple of times after tightening. 4 or 5 times should be ok. it's making belt to sit propperly
-check belt condition at regular basis. when removed, twist the belt inside out. if you see deep cracks between belt teeths, it's a sign the belt is at the end of a life

I hope this will help 

engine is gonna be in high revs, top speed will probably remain the same, belt wear will increase and mixture could be lean but uphill will be better.
higher revs require more fuel, so, smooth drive boss, 82 jet and then cdi cut. Just my advice 

It s not problem. I done some oil seals. problem are bearings. I got to buy bearing pullers to pull worn bearings out. thing is to heat case and to cool bearings in freezer and it slide in easily. seals are easy to remove and easy to install with socket, but need to put some grese on it to seal propperly