The part that gives me pause isn't that the employee is agitating for their own interest. That's good and expected. It's the method by which they're doing it.
It's OK if you and others don't think it's fair. We just disagree on that aspect. I don't think it's fair for an employee to use that particular method. To me, it comes off as a kind of extortion to the current employer, and mistreats the employer who made the offer.
Others can, of course, have a different opinion. Nothing wrong with that. We simply disagree.
To make a little meta-comment: I am genuinely surprised at the negative reaction I've gotten from a few people here. I really didn't think I was saying anything all that controversial. Live and learn, I guess!