Dunno, think you’re trying to eat your cake and have it too here…
> Mary’s Walk … [the] apostrophe … conveys information about the thing. Not that it’s important in this context, but in others, it is
The only information being conveyed here is a hint at the etymology, though, and as you rightly point out, that’s not important in this context, but it’s also sufficiently incomplete that the inclusion of the apostrophe could mislead. Was Mary a specific local woman, a reference to a biblical figure, a church, the surname of a local farming family, or a weird Northern word for something else? Is it a perversion of an old Danelaw word? Was it always a possessive name, or was that added by an enterprising local sign writer in the 1810s? St Albans is always written without an apostrophe or a dot, where ironically St. Kilda sometimes does have one.
I would suggest there are no situations in which this information is important to include in the name itself, and doing so adds a false layer of confidence that place and street names in the UK can be trusted.