So your link was for the special usage note to the last entry on the page that begins with "sometimes"? Furthermore I have never seen this idiom used when it is preceded by "can."
That looks like a suprisingly terse entry from the oed. I do not think io have eveer seen such a short entry from the oed. How much did you edit and leave out?
In the OED entry for the verb "speak," there is a sub entry for "to speak for." About half of that is devoted to "speak for yourself," which is what I quoted. It is followed by the typical list of examples.
There was a Simpsons reincarnation joke that used the phrase:
Sideshow Mel: You only live once!
Apu: Hey, speak for yourself!
"speak for yourself: expressing a desire to dissociate oneself from what another has just said or the assumptions behind it."