You can do it (worst that can happen is that you have way too much rosin on your strings), but there's not a lot of point to it.

I think the reasoning some people use is that they want rosin on both the strings and the bow hairs, so with new strings, they should speed things up by rosining the strings also.

The interaction of the bow hair and the string is what physicists call stick-slip motion. The rosin on the bow hair sticks to the rosin on the string, and pulls the string to the side until the force gets so great that the rosin can't hold any more, and the string starts moving. As it moves, it melts the rosin on the bow hair it's passing, and this goes on through several vibration cycles until the string has lost enough energy that the rosin can grab it again.

The melted rosin (even though it's only a few micrometers thick) will do a dandy job of getting rosin on the strings. If you do it that way, the worst that can happen is that you need to rosin your bow a little earlier. (With normal horsehair, that's like in 6 days instead of 7.)

In practical, everyday terms, I've never really noticed that there were any issues with the bow moving the strings when I put on a new pair of strings.  Getting enough rosin on the strings would take 3 or 4 bow strokes at most. And the bow will grip bare strings, it just takes slightly more force.

The only reason I can think of that you might want to do that is to make sure you have rosin on all parts of the string you might play on, so that you don't suddenly find yourself on a bare patch.  If you did it with the bow, assuming it takes two full bowstrokes (2 down, 2 up) times 5 soundpoints (the width of the gap between the bridge and the fingerboard is about 5 times as wide as the bow hair), times 4 strings, that would take 2x5x4=40 full bowstrokes.  Considering you'd need to do about the same to get rid of the extra rosin, I'm a little sceptical that it would actually benefit you.