We had an extended discussion through chat and solved your problem. This is a summary of findings and solutions that may be useful to someone in the future.
Although the Kohler 7000 engine in your Cub Cadet ZT1 mower comes with a low oil pressure sensor, Cub Cadet does not provide a warning light for low oil pressure as original factory equipment and does not shut down the engine if oil pressure falls below minimum. It was only by installing an aftermarket hour switch that doubles as a low oil pressure indicator that you learned that there was a problem.
The Kohler 7000 engine manual states that automatic shutdown for low oil pressure is only available on engines for stationary equipment that may run unattended, such as pumps or generators. Kohler does not support automatic shutdown on any equipment that is driven by a user. Cub Cadet runs a green wire from the low oil pressure sensor to a wiring harness inside the control panel, but does not connect it to anything.
The diagnosis consisted of three parts: (1) Is the hour meter displaying an appropriate message based on the input it is receiving? (2) Does the engine have oil pressure when running? (3) Does the low oil pressure sensor respond appropriately to oil pressure?
(1) The wiring diagram showed that the green wire that runs from the oil pressure switch to the hour meter should be a hard ground when there is no oil pressure and an open circuit when there is oil pressure. When you disconnected the green wire from the oil pressure switch, the low oil pressure message on the hour meter stopped displaying. Therefore, the hour meter is displaying appropriately for the input that it is receiving.
(2) You attached an oil pressure test gauge at the pressure switch port on the oil filter housing and read 0 psi with engine off and 33 psi with engine running. So you have oil pressure.
(3) You reinstalled the oil pressure switch and attached an ohmmeter to its single terminal. You read 0 ohms to ground with the engine off and it remained at 0 ohms with the engine running. The switch should have read “open” (infinite ohms) as soon as oil pressure reached 3-5 psi. Therefore, the switch is confirmed bad.
You replaced the oil pressure switch with a new one and reattached the green wire. The hour meter now displays a low oil pressure warning when the ignition is on and the engine is not running, and the message stops displaying as soon as you start the engine.