Any particular reason to think that Surrey has more non-Surrey children coming from outside the county to board in Surrey than there are Surrey children going to boarding schools outside the county?
I think it's likely to be the other way round, given that Surrey is an unusually wealthy county, likely to have an above-average proportion of children boarding compared to other counties.
There are 22 boarding schools in Surrey, which range from schools which are mainly Day like Priorsfield with about 60 borders to big names like Charterhouse with 800 borders. I'm not going to add up the numbers, but I'd give a finger-in-the-air estimate of around 3000 borders.
A lot of children will of course weekly or occasional board close to home. But anyway, boarding isn't actually big enough to be significant to these numbers anyway.
Even if all the Surrey boarders came from outside Surrey, and no Surrey children boarded outside the county - which is a ridiculous assumption - that's would still make it about 27,000 private day school secondary students compared to 60,000 state school secondary students.
Boarding schools barely make a difference to the impact.
And as for commuting across from other counties, there's likewise absolutely no reason to think that there are more kids coming into Surrey than going out.