Sounds like you have a good engine. Here is what I would do if it were me. I would pull the engine and tear it down. Then I would make sure to have all the bearings removed and just the bare block and crank in hand. I would boil them overnight in some antifreeze ( I use an old pressure cooker for this) while they are "cooking" I would meticulously clean the carb piston rod and sleeve. with everything good and clean I would give everything a rinse off with wd40 and a little fuel. Then reassemble. It sounds like a lot but you would know EXACTLY what you have and that it is clean. Now i also measure and check things but that is just me.
I would not change the engine, do not see the need for it. How does the rest of the car look? You said there were some missing parts. Is there any broken parts? Does it show signs of hard use?
I am the type that every used rc I get I clean the entire thing and check every part. When I get it back together I know what I have and that it should perform just how I expect it should. Dirt in the wrong place can cause a lot of headaches. Take your time and see what you really have there. You will probably end up with an rc that will last you a long time with proper maintenance and care.
Shaun