Car washes are an order of magnitude more water efficient (enforced by state agencies) than individual home car washing.
But car washes are interesting systems and one of the things that makes them interesting is that the water they use is (often) recycled. The same water gets reclaimed and re-used to wash cars over and over again.
I didn't look very deep at all, but one car wash chain suggests[1] a reclamation rate of 70%. Not perfect, but way better than 0.
You save water and electricity.
Given that by hand you use much more hot water than the dishwasher would, you're using more electricity (or gas). Since a little pump doesn't use much at all, it's mainly just the drying heat that matters. Though nobody is forced to use the drying cycle - if one enjoys drying dishes by hand, they can still use the dishwasher to do the washing.
eg. Our household conserves water (we track every litre we bring up to the tanks and filter) and we don't have a dishwasher - hot water is heated by the sun and run-off goes to the grey water tank for the garden.
And in your case, you use more water because you hand-wash, than you would if you used a dishwasher. You use slightly less energy because of the pump but also a lot more of your time, so... obviously how you relatively value those different things is up to you.
Our concern here, and that of many people in the area, is total water usage per household - and to a large degree water usage for washing, showers, and toilets is a small fraction of the total usage - and one that can be ignored as most people have grey water capture (as do well planned and infrastructured cities).
Water used inside the house runs out to water food, shade, animals, etc. and that use can be deducted from the water that is required for those uses.
Energy is no great cost, it comes from the sun - both as electricity via panels and directly as heat via solar hot water systems.
Is it safe? Not for a restaurant where the temps need to get to parasite killing temps to pass muster.. but if I had to guess, the odds of getting killed by a bad-tempered cow outweigh it by 10x.