That is your fuel pump. They can fail but they can also get dirt held up in them. From there it goes through a metal line to the carb which can also get plugged-especially w/ grass bits. If you remove the line end from the carb and from the pump you can blow through that with a compressor to ensure it is clean. Do not use the compressor through the pump. A clean piece of fuel line and mouth pressure is all you want through the pump. With a little probing, you should be able to tell if there is an obstruction or a failed pump. Ensure the fuel cap vents as well. Good luck.

Thank you for the advice. I cleaned the line and it started right up. I've noticed a new problem though, the throttle seems to be stuck in high. The butterfly things close when you move the throttle handle, but the engine never slows down, unless the blades are engaged. We don't really use it without the blades going, so it's not too big of a deal, but I'm curious if it's something we need to take care of or could just forget about. Thanks if anyone has advice about that.

On Kohler engines hard to see the actual butterfly in the carb due to the inlet elbow form the air cleaner. Are you watching the linkage or the actual lever on top of the carb that controls the linkage or inside the carb.???? or the spring could be stuck/dirty and not letting the carb linkage pull back closing the butterfly????

I would suspect or guess???? the butterfly in wide open and what you see is the throttle cable moving in the mounting bracket or the spring lever/ governor not pulling the carb. linkage back closing the butterfly. I would check the lock screw that holds the throttle cable tight so the rod or wire in the middle moves and the outside wrap don't. I have found it's really hard for a engine to accelerate very high with the butterfly closed inside the carb.

I would also clean the spring lever/governor where it comes out of the block using carb cleaner when the engine is cold and lub it with liquid wrench or something similar....

Are you confusing the "butterfly" valve of the choke for the throttle plate one?

I've had a throttle valve stick open from the shaft being gummed/varnished up, while sitting over the winter on a tiller engine.

You might manually rotate the throttle valve in the carb to make sure this isn't the case.

One other note, the metal fuel line from the fuel pump to carb needs to be replaced every few years. It is after the fuel filter so it will let rust into the carb needle and bowl area, bad gasoline starts the rusting process. It is a cheap part to buy. Bad gas or leaking needle can cause an overflow and stick you throttle.