Its just as important to get all of the silver salts out of the paper.
Too, a complex of silver-sodium-thiosulfate compounds form during fixing if the fixer, say, bath #1 is partially exhausted. If any of that junk is retained in the emulsion then no amount of washing will remove it as it isnt water soluble.
This is incomplete fixing and is most responsible for print staining over time. It might take years, but if it happens, the staining is permanent and cannot be corrected.
But, by following bath #1 with a fresh bath #2, these compounds are dissolved as they are soluble in fresh fixer.
Yes, of course the fixer must be well washed out to make the print as permanent as possible, but thats a secondary consideration.