I think it might actually help a little bit in the voting. You are identified as Mr. Astro, or Mr. Brewer or Mr. Padre etc., and people get to throw around words like loyalty and character etc as opposed to being veiwed as a mercenary.
I think that's right.
But more than anything, I think "nobody plays with one team anymore" is simply one of those classic canards, like "expansion dilutes talent," "players don't play as long anymore because of the big money" and "players are brought up younger and with less experience now."
No empirical support is brought to bear on the truthfulness of the statement, and even if it were, baseball pundits would continue to peddle the same lines regardless of the evidence. Of course, every one of these canards harkens back to the golden years of yore, when things were so much better.