I agree that cutting the whole limb is a bad idea; decay will most likely set in at the wound. Us arborists think long term, probably not a problem for decades. Comes down to if you value the tree long term or mainly see it as a nuisance.
Seems to me nuts will still get on the driveway even if you cut the limb; so maybe consider complete removal, or get someone interested in taking the nuts! In my town walnuts are on the "nuisance" list and you will get a permit to remove one in the City ROW no problem. Which is too bad, we have some 3 - 4 1/2 ft. diameter trees here (Black Walnut and California Black Walnut).
Value of the wood is mainly based on the diameter of dark heartwood in the logs. The butt chunk (can't see how tall it is -- if it is 8 - 10 ft. could be worth quite a bit -- big slabs with crotch wood can go for a lot --once processed and at a dealer, though). The large limbs (trunks) could be worth something, but likely don't have enough heartwood to interest a fine furniture grade walnut buyer. If they were part of a semi-load of hardwood, would be worth something, because lower grades with a lot of sapwood would get stained to be used for various products. By themselves, probably not saleable except as firewood (which is only worth something once in rounds on a truck). Maybe a local guy with a portable mill might want them and give you some boards in exchange.
I have dealt with a lot of clients (and potential ones) who thought that their not very big walnut was worth big money, when it was not. As in, we will make money if you remove our 18 in., branchy tree....not. OTH, there is a big one with a dead top in my town, nearly 5 ft. in diameter and no branch until around 20 ft. That butt log could be worth 10k at a buyer in Portland -- but you would still need to hire a 120 ton crane to put the 15k pound butt log on a flatbed in one piece and send it off (I think so -- crane guys here may correct me), plus the cost of a the crew taking down the rest, probably with the crane the first day to speed things up, and processing it (and it is over a street, powerlines, and a city park; probably two days, street closure and flag crew, probably 8k). So, the owner might clear 2k less the cost to haul.