This is part of the usual contract negotiation process, like an "infomational picket" in which the workers are not on strike but picket to bring the public's attention to the company's cheapness. No, most people won't care--unless they happen to be union members of some sort themselves, in which case they might. The purpose isn't to settle the dispute but to fire an early volley in what's usually a long a process.
In my experience, the contract provision usually said that no reporter's byline could be used over his objections. Mainly this was meant to allow the reporter to have his name taken off if he disagreed with major, substantive editing changes (and in practice, this type of protest is best used sparingly). By extension during a byline strike, all (or most) reporters simultaneously withhold this permission to use their byline. But a byline strike usually is not the major reason for that contract provision--it's to prevent management from forcing a reporter to take ownership of a story that, essentially, was not written by him.