I was cooking some food when I noticed a burning plastic smell and some smoke coming out of the knob, I let it cool down, and when I turned it back on, there wasn't any smoke but the burning smell returned, is the plate damaged, or is it some stuck food being burned?
The burning plastic smell seems to be coming from the plate, even when it is dissassembled like this, i can smell it from the centre.
That just looks like discoloration that sheet metal gets with repeated heating. I would investigate the controls, there are contacts that open/close as a bimetallic strip heats up and cools off (basic thermostat). if the contact are arcing or stuck it could over heat, also since you said smoke was coming from the knob. . .
I've been cooking some slow recipies that need a lot of heat (a lot of boiling water and slow stewing) under the kitchen fan, and the smell has greatly dissipated, it looks like the smoke was not a really big problem.
It is just a resistor, and appears fine based on the image. It was made in June of 2022, so it is not old. Most elements like this will smell when they are used for the first time. The resistive element is that wire you see sticking out at the white and red terminals. That wire is just a high resistance alloy and it sits in the center of a tube with a refractory compound that insulates it. So long as the outer tube or frame are not contacting the live wires, you should be fine. The wires appear to be high temperature with a fiberglass insulator and proper crimp connectors. The fasteners appear to have lock washers.
I can't tell for sure, but the component with the kapton tape on the opposite side is either a main high temp thermal switch, or a physical switch. The elements on the left and right appear to be adjustable thermal switches. That part might be interesting. Does this have both a high and low temperature cutoff adjustment?
I hope there is an earth ground wire somewhere in this thing, because that is the only thing I'm not seeing, but that will have no bearing on the smell/your present issue.
I've already used this hot plate regularly for about 2-3 months already. If this is burning off some other residue, how can I get the smell to go away faster?
The orange tape bit on the top is actually connected to a light that turns on when the heating coil is on. One of the elements is a power switch, the other is a temperature regulator.
Looking into the wires, I couldn't find a third wire, so it probably doesn't have a ground, even though it does have a 3 prong NA design.
I can't tell if that is a mica insulator on the bottom or galvanized metal. It looks like you could have spilled something on it.
With soldering iron tips and hot air rework stuff, I usually just cycle them to their highest setting a few times to get rid of any smell. I'll do it outside if it is particularly bad. I have a solder pot that is really bad and scary as hell when there are a few ounces of liquid tin/lead melted inside.
If you have a multimeter you can check the connection to be sure there is not a short between the chassis and element, but that is quite unlikely. If it has no ground, touching the thing would be... invigorating... if there was a short circuit. If there is an earth connection, it would trip your circuit breaker.
Mica is kinda a pain to cut and it makes a mess. It is a cheap high temp insulator. You should be able to buy a sheet or maybe even a disk that would require minimal rework to replace the one you have.
Personally, I'm cheap and would just head to a thrift shop. Most consumer junk with a heating element will have a mica insulator sheet or few. Something like a $1 toaster should have something you can physically see without taking one apart in the store. Most cheap toasters have mica sheets with some ni-chrome wire wrapped around the sheet(s). It is probably fine as is, and the smell will likely dissipate in time.