Well now! I'd say that "the road" is "public property," and since, clearly, the city is not going to be taking care of the ground cover that's growing on their Public Property, that you'd be "rescuing" it!

Seriously, in the situation you've described, I don't think there'd be any moral dilemma at all about pulling out a couple small starts for yourself. Clearly it's able to spread by itself and will fill back in any small spots where some has been removed, so I don't see how you'd be "damaging" anyone. Do you have any idea what it is? It's possible that it won't die when it gets hot--if it's spread that far on its own anyway. Possibly woolly thyme or something that works really well in such a situation. Or some sedums, like Sedum album will spread anywhere too. It would be good to have some idea what you're getting just to be sure you're not getting into something invasive (like S. album!), that you wind up trying to get rid of in the future!

I've been known to "appropriate" a hen & chicks "chick" on occasion to help "improve" my "collection!" CLEARLY I've never taught ethics!!!

Skybird

P.S. If you decide to get a start of whatever it is, I recommend doing it NOW while the soil is wet, or you may not be able to pull any out with the roots attached!

EDITED to add: I agree with popmama if you do it and you're going to wind up wishing you hadn't. I almost put down that you could just ask them what it is and say that you really like "what they've done" with their yard, and could you remove a small start for your own yard. Doing it that way you could find out what it is--if they know. For myself, in this case though, since it's growing all the way out to the road I don't really think there's anything wrong with just removing a small start. Just depends on how you're gonna feel either way.

Now is definitely the time to pick if you are so inclined. The ground is practically giving plants away!

I just did it. At night. Furtively. Already planted.

Thanks Skybird and popmama for "moral permission," as they say. :)

You have me curious, so if you figure out what it is, let us know!

Happy (furtive!) plant acquisition,
Skybird

I'll take a pic in the morning for you, Skybird. I have no idea what it is.