Uranium recycling is not the same thing as breeder reactors. It's essentially the same nuclear enrichment we do to natural uranium to bring it up to concentrations of U235 as usable fuel, just used on spent fuel rather than virgin uranium.
Also, for uranium sea water extraction the plan is to drop buoys with the absorbtion material and let natural currents bring water into contact with it. Pumping all that water out of the ocean would of course be stupid.
Recycling: however breeders are AFAIK the most efficient way to tackle this (however it is so difficult there is no adequate industrial reactor, after ~70 years of prototypes and research). Other ways don't seem very appealing, for example France ceased to recycle in 2013.
Yes, since the 1980's a fair amount of Grand Plans aimed a extracting uranium from seawater. Nothing industrial yet. I won't hold my breadth.
Uranium isn't recyclable: there is no satisfactorily running fast-breeder.
My point was not about costs but about emissions. Extraction and immediate post-processing (before enrichment), and therefore ore grade, have a major impact on emissions: see figure 5 in the referenced communication (Werner, Heath).
Seawater: "pumping the seawater to extract this uranium would need more energy than what could be produced with the recuperated uranium" Source: http://large.stanford.edu/courses/2017/ph241/jones-j2/docs/e...