Faulty valve cover causing fuel pump issues? (Kohler SV710 Twin Courage 20HP)

I have a Kohler SV710 Twin Courage 20HP engine on a riding mower. As part of spring tune up, I recently replaced the stamped metal valve covers with plastic ones, since I had to check and set the valve clearance. I did the additional regular maintenance (oil, oil filter, OEM fuel filter, spark plugs, and replaced all fuels lines, from the tank to the carb, with reinforced lines purchased from a local mower repair shop). The engine started fine, ran fine for a few seconds then started to hunt.
 
I thought dirty carburetor. As I am new to engine maintenance, I decided to replace the carb with an after-market one (not being confident that I could clean the OEM carb without messing something up). Same issue. Engine starts fine, runs for a few seconds at high throttle then starts to hunt. Since then, I opened up the original carb. It was surprisingly clean; not a single jet clogged.
 
I finally noticed that the fuel filter, starting half full, empties after the engine starts and remains almost empty. The engine appears to be sipping fuel, hunting, until the RPMs eventually even out. Never had that issue before. At idle, the engine hunts without stabilizing. (I have not been able to test the engine under load, but suspect that it will bog down when cutting grass.)
 
I suspected the fuel pump. But the fuel flow out of the pump seems reasonable (I have no experience with this). The flow of fuel from the tank to the pump inlet is a heavy steady steam (gravity feed). Nothing is clogged. The fuel tank is clean; no debris at the bottom near the outlet. Yet, it appears as though the fuel pump can’t draw sufficient fuel into the filter. I changed the fuel filter with another OEM. Same issue.
 
I am now starting to think that the plastic valve cover and the new pulse line are somehow defective, causing insufficient suction when the pump is pushing fuel into the carb. Does this make sense? Can a pump be weak?
 
I have not run the lawn tractor for a long time since the tune up. But I have been working in it over the weekends for 3 weeks. There is no oil dripping around the new valve covers. The O-ring seems to be sealing well. Other than changing the fuel pump, and/or pulse line, or re-installing the original metal valve cover with RTV (yuck!), I am not sure what to do next or whether suspecting the fuel pump makes sense.