Pointers for cleaning and re-finishing a brick chimney interior

I've tried google/youtube and can't seem to find any decent tutorials/instructions for my particular situation, so I'm hoping someone can help get me started on this repair job. I don't necessarily need a full breakdown or step-by-step instructional, but some help with the terminology, techniques and materials I would need to learn and use to accomplish this project would be greatly appreciated!

The project is the interior face of a very old chimney that protrudes through the floor of a 3rd floor bedroom and up through the ceiling. I believe the term for the stuff on there now is called "render" as it is like a stucco/plaster smooth-ish finish nearly an inch thick overtop of the original brick. This finish coat is in bad condition and falling off the brick near the ceiling. The rest of it is still holding but there are some cracks and I'd rather not test its integrity until I know what my plan is. The chimney is lined already and is used exclusively as a vent for a propane boiler in the basement. There are no fireplaces or anything in use anymore. This is in an early 19th century farmhouse in the US, so the brick is very old but still solid (no loose bricks or anything like that). Bits of the mortar are loose and will likely need to be re-pointed if I take any of this stuff down but I'm happy to do that.

Ultimately I just want it to look nice, not be falling apart and creating dust, and probably most importantly: not leak any rogue gas fumes into our soon-to-be guestroom. I think because the chimney has a liner in it already, the gas isn't an issue but I didn't put it in so I really can't be certain of what kind of job was done. I'll put a CO2 monitor in the room regardless, but would rather avoid that problem altogether. I'm fine with either reinforcing the render to give it a few more years or cleaning it all off and resurfacing the brick somehow. Any suggestions?