In the second picture, the linkage is the one closest to the engine shroud.

Thanks for the reply and sorry for my delay. The 16HP Power Plus (Briggs) engine model number is: (Model, Type, Code) 286H77 0121E1 040123ZD Thanks, Nick

The "simplistic" alternator check function on most consumer grade battery chargers will not recognize the weak output of some small engines. Best to carry that battery (fully charged) to an auto shop or parts house and let them load test it. Testing the output of the engine's charging system can be done easy enough, once you get the engine started. 12.3 volts IS NOT a good reading to have. Ideally it should be 13.2, or even 13.6. Once you get the engine running, with the engine running at wide open throttle, a volt meter should register about 14.2 or 14.4 volts, after it has run at that speed for a few minutes (to put back what was sucked out by the starter). If your system voltage DOES NOT reach 14 volts, you have a charging problem. We can cross that bridge when we get there.

the 200 series IMO are decent machines, not up to the 300 series for sure, but still good. The price is right thats for sure. I would only put a new engine in it if the rest is mint. You might be able to free it off. Take the head off and apply oil to the cylinder. Gently tap on the top of the piston with a block of wood and see if it moves. If the bore is scored up its a goner. If it doesnt move with taps from the wood, its probably a goner. Best idea would be to drop in a good used engine.