It's a misguided attempt for cities to feel like they are responsive to that portion of their citizens who want their city to support this kind of thing, without the city having to really do anything.
I have a clover lawn, and love it. I also have invested heavily in native plant gardens which I see as havens for the pollinators and native species who have little habitat remaining.
I think that this effort would bear far more fruit if it involved a component of 'planting native' in your yard for pollinators. Clover is great as it is more drought tolerant, does not need as much fertilizer, and simply looks better to me. But it does not really support native pollinators; if anything just European honeybees which are their own issue.
I do appreciate that it has given some folks more latitude in establishing a non-traditional yard, and hopefully is shifting attitudes away from the monoculture lawns of the past that suck up chemicals and kill pollinators and birds.
No Mow May is just a gateway drug for native plant enthusiasts, I hope.