Well, sorry, it sounds like I'm unable to help you with something that fully meets your criteria.
To be clear, I'm not planning on harvesting anything next summer, but something will of course be picked if there's time and energy.
Soybeans might be a possibility. But if we do it's not something we want to regenerate. Or more precisely, is it something that might become invasive? How often would they need mowed?
Soybeans mature in 2 + months...you stated "grow from last frost to first or longer is what I'm after"... obviously to exist from last frost to first frost, regeneration is required. In your area, you are looking for cover May, June, July, August, Sept, and part of Oct., right? Beans, soybeans, cow peas, etc. are not realistic for that time period without regeneration.
Sweet potatoes aren't something we eat much. I had thought about regular potatoes but didn't think the greens would get dense enough. How do the 2 compare?
This is confusing "To be clear, I'm not planning on harvesting anything next summer" followed by "sweet potatoes aren't something we eat".
The reason I suggested sweet potatoes as an option to consider was 1) they will grow and flourish May, June, July, Aug, Sept., Oct. Last frost to first frost which meets your criteria. You can eat them or not, does it matter? They grow thick vines which choke out weeds...something else you were after. They require no maintenance...something else you were after. They are not invasive...although it would probably be a good idea to dig them up after first frost to prevent any residual next spring.
No, they are not at all like potatoes which fail to meet your criteria completely.
Regarding buckwheat, is that something that might become invasive? In other words, if I mow and till the following spring is it something that I'm going to have to constantly pull out?
I'm a little paranoid about introducing something I can't get rid of. It seems we've finally turned the corner on the chickweed that came via manure.
Buckwheat is not considered invasive...but it will go to seed in about 12 weeks if you do nothing and can come back from the seed the following spring. So,
rule it out unless you are planning to mow it before seed maturity and reseed, regenerate it at least once during the summer.
Fwiw, I intend to do the entire garden, including footpaths.
I tried...maybe someone else can help you.