Heh, you hit my specialty.
The size of those filters (claimed at 15 nm) is greater than the size of most solvated ions (some, like SO4--, may be larger, but something like 88 wt.% of seawater salt is on average sodium and chloride ions). It is, at the moment, very difficult to consistently manufacture molecular sieves smaller than 30 or so nm with traditional phase inversion techniques and probably smaller than 20 nm with other techniques, which is why I'm a bit skeptical of the manufacturer's claim above. But let's give them the benefit of the doubt, because even with a molecular sieve that filtered out salt ions (around 1-1.2 nm) you would not be able to draw water off that bottle.
The energy cost of desalination is pumping--you have to pressurize the liquid above the osmotic pressure of water to pass it through your membrane. The osmotic pressure is proportional to the concentration of everything that resists movement through the membrane (i.e. not water but all the contaminants), and when you do the math for seawater you usually have a minimum applied pressure of somewhere near 25 bar. Actual RO plants usually operate above 50 bar, sometimes as high as 80 bar, for efficiency reasons. Long story short, that kind of equipment as well as doing proper cleaning and antifouling is expensive and it's been easier in the US to use nanofiltration on freshwater and wastewater resources rather than go all-in on RO.