There is absolutely no advantage in using synthetic oil in an old engine designed to run on normal refined oil but it will not hurt your engine.
Just try to keep within the recommended viscosity ranges.
Using a 5W 40 synthetic in place of a 10W 30 will cause a little bit of oil smoke on start up and might leak a bit more past the seals when the engine is being stored.
But when running, no difference.
If it floats your boat then go ahead.
However you still need to change the oil at the same times.
Don't get sucked in to the doubles / triples the oil change intervals.
That only applies to auto engines, not mowers.
When I worked for Mobil, back in the 90's, people would come into our place and we'd sell them Mobil 1. Even though they'd been running regular oil for ever. A lot of them, within a couple of weeks, would come back in with oil leaks. Simply because the synthetic would clean off all the dried up oil that was keeping it from leaking in the first place.
From what I've learned, oils with plenty of zinc in it is good for older engines. I'd like to know what brand or oil has plenty of zinc in it. Kawasaki labeled oil does. But that's hard to find off the store shelves.
Project Farm did that Oil showdown series on youtube.
He sent oil samples off to see what and how many anti wear additives they had.
I think Valvoline and Castrol had the most zinc, not sure.
I know that Valvoline racing oil has more zinc than standard oils.
And kinetix brand oil is made specifically for air cooled engines.
Kawasaki oil (K-Tech) is made by Citgo.
THE ADVANTAGES OF USING OIL WITH HIGH ZINC CONTENT
Despite the fact that boron-based additives have come to rival some zinc motor oil additives since the limitations imposed by API on zinc and phosphorus usage, the latter are still the best when it comes to dry start protection and a host of other advantages. Zinc and phosphorus are added mainly to improve surface protection, having the important role of protecting major engine components from wear and oxidation, and removing the likelihood of solid deposits. Zinc is also known to offer extreme pressure protection, and to protect components made from ferrous materials from corrosion. As the oil thickens as a result of extreme heat, it gradually thickens, forming deposits of varnish which prevent oxidation from even occurring.
ZINC AND OLDER ENGINES
Most engine manufacturers have a minimum requirement of 1,200 parts per million of zinc when it comes to the engine oil you use. In fact, some will go so far as to void your warranty, if that amount isn’t found on oil samples taken from broken engine components. When it comes to older engines and classic cars, high zinc content is a must, and some oil manufacturers even sell special engine oil that comes with elevated ZDTP or ZDDP (Zinc Dialkyl Dithiophosphates) content. ILSAC multi-viscosity oils that are rated SM even feature special additives used for older engines that experts recommend for use in older engines.