Problem is:
Dirty laundry can be stored practicly anywhere. It might smell a bit unclean after several days, but put them away in a bucket, basket or even a seperate place in your closet and nobody will notice them.
Then, every week or 2, you take a whole bunch of laundry to the laundromat. As they are easyliy stored in a bag or basket, transporting is easy and they don't weigh a ton.
With dishes, you'd have to store them somewhere as it is unpractial to go to the "dishmat" after every single meal. The only place I can imagine storing dirty dishes is - guess what - a dishwasher.
If you try to get it somewhat economical for a customer to use tha service, they would have to store their dishes in still dirty condition for 4+ days. That not only smells, but probably is more unhygenic then any sponge.
Now you have to transport those heavy, fragile items several miles probably. Without getting any soil of your dishes onto you or your car seats\ on the bus seats.
Further, DW need more attendance then washers. Most German laundromats I know are controlled by their owner. They come arround once a day (usually in the evening, before rush hour) to check the store and get the money. Once a week, they vacuum and clean the floor, clean fluff filters (some do this every day) and clean washers.
A commercial dishwasher needs thourough cleaning every day (checking spray arms, cleaning filters etc.) plus is more demanding in terms of chemicals (if a laundromat user overdoses, the washer usually just completes it cycle with badly rinsed clothes; if a DW is run with hand washing detergent, it usually ends up as a pretty big mess).
I doubt there is a high demand for such a facility. Especially with compact and mobile DW.
Oh, and 1$ is probably beyond doable for the owner.
Funny your profile mentions you have 2 DW.
If you use this much liquid detergent for hand washing, you are doing something wrong. At least if I got you rigjt and you use one bottle of detergent for one wash up. We usually need about a bottle per month.
Usually, I start by pre scraping everything. Pans and pots and such get a short rinse, are then (if soiled) filled half with some water and bit of detergent and set on the cooktop on lowest heat to soak. If the water starts steaming, turn them of.
While they soak, you fill the sink with enough hot water and a teaspoon of detergent usually. Now, wash the dishes, cuttlery etc. in that water. Start with plastics and glasses and cups (usally less soiled) and move on to plates, cuttlery and other more heavy soiled things. So, work from low to high soil level.
You can rinse them if you feel the need to, but usually that is not necessary.
After you finished washing all the dishes, grab the things you set to soak. Pour the soaking water into you washing water first so you don't hurt you self and then start scrubbing them. Usually, the hot water and high concentration of detergent work really well and things are easy to clean.
Of course it takes long, but nothing unsual in most parts of the world.
And if you are scared by dirty sponges and clothes: Wet it up, put it in your microwave for 1 minute and you should be mor then safe. Works with woodden cutting bords (or cutting bords in general) as well: Wet, microwave, rinse.
My suggestion to you would be a compact countertop unit. They can be usually carried by one person fairly easy. Store it in a cupboard or closet, get it out when needed, hook up to you sink. When done, drain and store again.