I don't know of anything that exists like it, but perhaps someone out there will. Is there a pedal system and shoe, or simply a shoe that allows the cleat to be mounted way behind the ball of the foot so that it is nearer the arch of the foot? All winter I rode my cruiser with the ball of the foot over the pedal and found it much more comfortable for my knees and when I jumped on the road bike today I thought how much better it could be with a cleat mounted closer to the arch.
Any ideas?

Pretty much have to drill & mount yourself AFAIK, although I think speedplay has a plate that gets you partway there.

More info:

https://www.stevehoggbikefitting.com...leat-position/

Can't help with specifics, but I remember talking to Lynette Chiang (https://www.galfromdownunder.com/) about having the cleats back as far as possible. She said she had modified her shoes to put the cleat in about the middle of the sole instead of the ball of the foot. Also said that Lon Haldeman (long distance racer) had made similar modifications to his. Sounded like she hadn't had much trouble making the change to her shoes by drilling new holes and AIRC she was using Shimano SPD (mtb style) pedals.

Are the cleats on your road bike shoes already adjusted as far back as possible? If not, perhaps you can try that first and see if that's already good enough, or if you still need more.

Last I looked into it the speedplay-specific 4-bolt shoes allowed for the greatest movement toward the midfoot but the shoes alone are ~$500 so... I just opted to stick with XL toe clips which may get your food back about as much as you want.

I prefer my hybrid with platform pedals nearly centered under the arch of my foot, but when I switched the road bike from platform to clipless Look pedals I was satisfied with the Delta type cleats as far back as they'd go with the existing holes in the Scott shoes. Not quite centered under the arch, more like midway between the arch and ball of the foot. But comfortable enough that I didn't feel any need to readjust them or drill additional holes after a few months riding that way. It worked the calf muscles more for awhile but never felt like it was straining my knees, etc.

100% on that - i had a costly crash due to trying out a set of new Bont riot which somehow didn't allow me to move the cleat forward enough....the front tyre caught my left foot and i got stuck - over reacted and fell over.

I guess if you only rode roads then toe overlap may be a small issue - but i'm riding a lot of bicycle paths which have lots of turns - some with 180 degrees....so these are tight turns. Toe overlap can be a serious issue. My confidence took a hit on turning these days with toe overlap.