The trick is to reuse the water for scrubbing/soaping and "flash rinse" by quickly turning off the water when rinsing.
Also, that's not how I wash dishes by hand any more; I splurge with the warm water used per cleaned item. But I still don't let it run for more than a few seconds. It just feels wasteful.
> you’re still warming up the water somehow
This is somewhat a sunk cost for most people: Unless you turn off the warm water in your pipes, constantly keeping warm water available costs something and using a bit of it costs very little in addition. We really should be thinking of turning off water heating at times when we know for certain it's not being used, and only heat up water anticipating usage. There's a lot of energy to be saved for modern smart homes (or anyone who cares to manually regulate, or simply has to because they're using wood burners and such).
That is assuming you don't use highly isolated water bodies as thermal batteries.