Well, it is perfectly possible to restart browser integration on spin change, but it is "Default port", not actual port, because if the default port is already used, Picard will try next one (ie. 8001).
I think it isn't desirable to have it change on every spin change (ie. one spins from 8000 to 8010, we don't want it to restart at 8001,8002,etc...).
If one wants new default port to be used right away, then it is simple to click checkbox twice to disable/enable browser integration.
You should try it, i think current behavior is ok.