It is the fuel, and almost everything else follows from it.
What happens chemically. Petrol is not stable. Over months the lighter fractions evaporate and the rest oxidises, leaving behind gums and varnish. It is also hygroscopic if it contains ethanol, so it draws moisture out of the air, and over a winter that water separates out and sits at the bottom of the tank and the carburettor bowl.
Why the carburettor is always the casualty. The passages in a small engine carb are tiny, the main jet is a hole you can barely see. It takes very little varnish to block or restrict it, and that is exactly where the residue ends up as the last of the fuel evaporates. That is your third year in a row, and it will be a fourth unless the storage changes.
Why it is worse in a dry shed than you would expect. Temperature swings in an unheated space push air in and out of the tank every day, and each cycle brings in new moisture. A sealed heated space would be kinder, though almost nobody has one.
The other two things that suffer: oil that has absorbed combustion by-products becomes acidic and sits on bearing surfaces all winter, and a battery left flat sulphates.
So the October job is fuel first, oil second.