Threads about oil choice are as pointless as discussions about politics and religion. There is no such thing as a single answer.
Your tractor has a Kohler KT-17 engine and hopefully it is not the dreaded Series One, which was a problem for many owners. If you want your engine to have a long life then put an hour meter in a convenient spot so you can track the actual amount of engine use. Then, create a log book so you can record the date, engine hours and what service work you performed. Keep a file with all the invoices for parts you purchase for your tractor.
Use a good quality motor oil, as per what KOHLER themselves tell you to use. Most likely, this will be straight grade number 30 detergent base oil and then change it every 25 hours maximum. What matters most to your engine is CLEAN oil that is always at or near the FULL mark on the dipstick.
Change the air filter on a timely basis and keep the engine tuned up once each year. Make sure the cooling fins on the block are kept clean and that nothing is blocking the air inlet on the front of the motor. It is important to keep your engine from overheating. Using a synthetic oil can help MARGINALLY with keeping the block temperature down but synthetic oil is not some sort of magical potion that can take the place of proper maintenance and timely oil changes.
Synthetic oil gets just as dirty as regular oil does. The only thing that helps dirty oil is an oil filter and the KT Series engines don't have one. One of the big advantages touted for synthetic oil is that you don't have to change it as often but that only applies to modern engines fitted with an oil filter. If you want to spend the extra to run synthetic, then by all means, do so as along as you adhere to the 25 hour oil change procedure called for by the manufacturer.
I don't know what make/model of hydro that is in your tractor but if you join Red Square, which is the dedicated forum for Wheelhorse tractors, you can find out there. See the link below.
Some hydro's come with an oil filter but many do not. Once again, the oil is the lifeblood and should be changed on a regular basis. We are only talking peanuts here expense-wise for the oil and it is always better to change it too often than not often enough.
Anyone who tells you that synthetic oil will extend the life of an old KT-17 engine, is doing so with absolutely no basis in fact. If running synthetic makes you a happy camper, then use it because it can't hurt to do so.