Hello, great of you to produce the video, excellent!
Yes, you can take care of this without any trouble or special knowledge.
I'd say, take a look at this thing, here an ET 4 125 motor with the variator cover removed. The red line shows the starter gears;
You'll need to block the variator wheel from turning, this tool is great, costs around EUR15. Here, the green line.
Once blocked, the wheel is removed, it just pulls off, blue line.
The small gear assembly just pulls out. Clean it, re-lube the gears, make sure the spindle moves freely.
No cost as yet, this may do it for you.
Otherwise, I'd get another, or a new starter. The Vespa is a 2005 model, a starter motor is not a great cost factor.
Remove the carb, and you can access the starter motor from the top.
Welcome to the forum.