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Sounds to me like your starter motor has brushes that are worn out or sticking. If you can tap on the motor and get the starter to work, the brushes are sticking and not making very good contact with the armature, or the springs that hold the brush up next to the armature are weak or broken, or the brushes are worn down so short that they no longer reach the armature.
I would open the starter up carefully(watch for flying parts), check the brushes for condition, make sure they move freely in their holders, clean the armature contact area with a toothbrush, blow out all debris from the armature and field windings, lube the bearings.
Brushes can be replaced, if you can find them, Certainly cheaper than a new starter,

I have a PDF copy of a service manual for the NC50 if you are interested in working on it yourself. Let me know.

The 50cc bikes are usually governed so they stay at or under the 30mph speed limit that most are designed for. The CDI has a rev limiter built in, so you need to buy an aftermarket CDI with no rev limiter in it.

Mark Twain. Runningslow, welcome to Texas.

Metal stud framing screws work pretty well for that, too.

I run relatively cheap Shenkos on my 250 and they have held up well in the rain--of course I don't drive like an idiot in the rain, either. Plan your moves and don't do anything sudden, especially hitting the brakes.

Now if you want to drive in the snow, check this out:

www.riminimoto.net/prodotti.asp

Now this looks like fun!

I watched your video and it sounds to me like you might need to clean or adjust the carb. It seems to be hesitant to rev up when you give it the gas. Does your temp guage read hot? I couldn't see it well in the video. It may be running too hot, making it hard to crank. When mine gets hard to crank, it is usually time to change the oil. When it gets old and thin, mine gets hard to start.

The start and charge systems on these bikes are barely adequate when everything is working good. I noticed you kill and restart about a half dozen times in a few minutes. You need to give it some time to recharge the battery when you start it--at least a few minutes, or take it for a short ride. Idling, with the lights on full time, will probably not charge the battery much, at least not very fast. Your meter showed it to be charging, but likely not very many amps. You need to ride it to get a decent charge.
Looks like you have it set to idle around 1000 rpm. Can't tell too well on my monitor. I would idle it up around 1800 to 1900 rpm or as high as it will go without engaging the clutch. It will help the charge and might help it to crank a little easier, too.

Does yours have the remote start? I kept having battery problems until my dealer told me about the remote start. If you start it with the remote, kill it with the remote, or it will still be "on" and it will drain the battery.

I listened to the video again, and it does sound to me like the bike has a miss at idle, and sputters on rev like it is dirty/running a little lean at idle/off idle. Clean or adjust.

You could probably go for weeks or months without a muffler (I wouldn't) but you won't last but minutes without oil.
I make checking the oil part of my daily walk around before I crank it up. (Especially if I'm riding someone else's bike.)
Like jct842 said, you won't know until you open it up. My guess is, at the least, you are going to need a piston, rings, and jug. If it seized while hauling azz, you probably snapped the rod, too, meaning a total rebuild or replacement.