This machine is so annoying… Why do I need to manually tune the z-probe height every time after auto-leveling completes?! Is there a better way to do this?
It depends.
Why do you have to tune?
If the nozzle touching the washer at any corner results in ANY visible movement of the bed, your nozzle is not clean enough for probing and the tuning you’re doing is to compensate for the inaccurate reading from the dirty nozzle. Ensuring the nozzle is clean is a must, until you give up and install a BLTouch adapter and customized firmware to use that instead of the washer-based system.
If there is no visible movement, and you’re pushing in the wheel to do the live Z-height adjust during the first layer, then you just need to change your z-probe offset in the menu, and save your settings.
There’s also a chance you’re operating with some modified startup GCODE that manually set the Z probe offset, but that is unlikely unless you’re using a public printer (or you did it yourself and forgot you did that. That happened to a um, friend of mine.).