Teachers rarely shop around at all, though when they do it is most likely for reasons other than pay like a better faculty fit, improved schedule, or better-aligned educational ideals. For teachers I know, which span both public and private realms, pay isn't a highly motivating factor, whereas the items mentioned above are.
In fact, many teachers at private schools are paid far less than those at public schools - even up to half as much. Why? Because they believe in private schooling and enjoy the freedom that comes with it. Oddly enough, many of these private schools with low pay have much higher performance than the surrounding public institutions.
I have a theory that student success is much more tightly aligned to parent involvement than that of whether teachers are highly paid or not. In private schools, parents pay for the education directly out of their own pocket and therefore have an expectation to get their money's worth and to not have that money squandered.
But you are right that most teaching positions do come with great time off and benefits - even for those starting out.