Oh good, autorefresh is turned on.

I had a reasonably coherent reply typed out regarding that while I have a theory I'm not convinced the hydraulic fluid would ever get hot enough to transfer enough heat through the manual extension handle to get it up to a temperature that would cause burns like you described. That and the Cessna RG system on their singles sucks
A leak at the power pack which I believe is mounted inside the cabin between the two sets of rudder pedals provided it's an electro-hydraulic airplane wouldn't necessarily have any hydraulic fluid getting ahead of the firewall but a leak at the nose gear actuator or associated lines/fittings certainly could. A GoPro shot of the ammeter during your flight and hearing the hydraulic pump kick on as it constantly tried to rebuild the required pressure to hold the gear up would be telling. The gear on the 210 like other Cessna RG singles (the system sucks) is held up by hydraulic pressure alone rather than mechanical uplocks. If a leak were to develop in the system resulting in a decrease in pressure the pump would kick on to recharge the system to whatever PSI is required to hold it up (750? It's been a long long long long long time) which is normal enough as the pressure bled down over time. The pump kicking on isn't anything that would cause alarm but should a leak develop or the pressure switch fail (this would confuse the pump into thinking the system either is fully charged all the time or always low) you could have the power pack running much more frequently than usual. There's no question it gets hot if you do this and can have the CB pop if you're doing repeated cycles of the landing gear on a hot day such as touch and go training as it's happened. Would it get so hot it burns your hand? Don't know, never tried it. At any rate, if a leak were to develop which resulted in the pressure being low enough to warrant the hydraulic pump to run continuously consequently heating the remaining fluid up to whatever temperature is required to transfer enough heat through the manual extension handle to burn your hand you're actually lucky to get the gear down at all. If the airplane was successful in pissing away all the hydraulic fluid you'd have been left with no means of extending the landing gear because the fluid you have in the system is all you've got to work with. Gravity or free fall doesn't work, the theory of using a tow bar to grab the legs doesn't work if you're solo and frankly the risk far and away outweighs the minimal hazards of a belly landing. Especially if you only were to get one leg to extend.
Not being there to smell the smoke doesn't help but it's intriguing the normal extension didn't work but the manual hand pump did over a hydraulic problem. They both do the same thing and unless the smoke was from the power pack motor burning up as a result of the separate hydraulic leak causing it to run/overheat/let out the magic smoke I can't figure off the top of my head what causes that same combination of failures and allowed the smoke to stop generating. At least not at 2am.
How frequently was the power pack charging the hydraulic pressure? Was the power packs CB popped? Was the smoke smell electric or was it the smell of hydraulic oil?