So you have the mechanical pump integral with the left rocker cover ?
Are you sure you have it hooked up right way round ?
Check by cranking the engine with the carb end of the hose higher than the carb,
Should shoot fuel at least 2" higher than the height of the carb inlet.
After that it sounds like time for a rebuild kit in the carb.
The single barrel carb is prone to hardening of the float which will develop a crack thus no longer float.
The old pump could have allowed bits of rubber through the fuel lines which are now blocking or partially blocking the fuel inlet.
Please take photos of the carb linkages as Kohler have a single throttle plate which is used for every application thus it is very confusing when you go to hook it back up again,
Which wire in what hole & where did that spring go ?
Kohler service manuals are available from Kohler direct as a free download and if wort comes to worst they are all on the K & T web page
https://www.mymowerparts.com/pdf/