Actually, it's the other way around. By building Light Water Reactors (burners) instead of gas cooled fast reactors (breeders), we would now be out of uranium. Light Water Reactors do produce plutonium, but it is impossible to extract  at a reasonable cost with today's technology, let alone the technology they had back then.
If we do manage to invent a profitable way of extracting plutonium from "fissium" for fabrication into Light Water Reactor fuel, we are in good shape. Otherwise, we can still build gas cooled fast reactors with the fuel we didn't burn in the hundreds of Light Water Reactors we didn't build.
But global warming wouldn't be as bad as it is now, that's true.
It is illustrative to remember the problem isnt closed fuel cycles are so expensive, but uranium is so damned cheap.
I worked on the GCFR a little in the late seventies/early eighties, when support for the HTGR ran low.  Did you? at GA?

The latest greatest is liquid metal reactor pushed by argonne that burns light water waste (with cadmium and short lived bits removed), essentially burning all the actinides.  This might be the way to go if/when uranium gets dear.  Meanwhile, hold on to that waste... it will keep our kids out of the dark.