For mowers like this, I have found two types of oil to work wonders. Either things greatly improve or the fact that there is no longer TAR in the engine will allow a seriously worn engine to flow apart within only a few minutes.
For the more low-tech flathead engines that generally suggest 30W or 10W30, I have standardized on a Rotella T6 15W40 diesel rated oil. This has greater anti-wear additives and will not be of concern with fouling emissions equipment. Flathead or L head engines are more common on single cylinder push mowers but can be found in some older riders and other larger engine equipment as well. I had been using the Rotella T6 5W40 for a while on some of these and noticed more oil consumption and blowby as mentioned. Switching to 15W40 greatly helped with this. Most of this is run in higher temps outside as well so never sees winter use.
For engines used in the winter or OHV, I use the Rotella T6 5W40. Generally these specify 5W30 or 10W30 from the factory. Again, I have seen this work wonders or result in the fly apart of a seriously worn engine. It is either one or the other.
One success story with this was a rider with the Kohler Command commercial engine. It had been seriously neglected and the oil never changed. It would just blow smoke and bog down at the first hint of resistance/tall grass. I put the 5W40 in it and mowed once. By the end the mower was way up on power and no longer smoking. I drained it again, changed, and ran for an entire season before changing. The oil didn’t look bad even then. This also used hydraulic lifters which were super clattery at first. That problem also went away. I ran it for several years before the rest of the machine was pretty much falling apart and more of a pain than it was worth. I had been beating the crap out of it cutting brush and such once I had to start it with a battery right to the starter terminals. I sold it for $50 with a lot of other mowers and junk I was moving along for parts, scrap, or whatever.
Generators get 5W40 no matter the engine type as I figure they will be more of a winter item.
I never re-use oil with a few exceptions. I have used good used oil that I know doesn’t have contaminants to flush seriously neglected engines out. I run it 5 minutes, then drain. Another time I had a Kohler Courage engine on a rider. Then it decided to start acting up in several ways and one was burning about a quart of oil every 10 minutes of running. It would even start dieseling at times with flames out the exhaust from all the oil. It would run with the key off like this and even shutting down with the deck engaged wasn’t enough to get it to shut down at times. It would just keep running on all the oil it was burning. Starting it was hard and of course you had to walk away and let the smoke clear. Once it got running, the smoke turned to fire coming out the exhaust. It didn’t always do this with the deck engaged but once you disengaged, it would start to rev up and blow lots of fire at times. Again, this one got moved along after not too long.