The thing is...hunts are supposed to be hunted. It's something you are intended to actively pursue. Instead, most people just wait around for someone else to shout it out and then want them to wait however long it takes for the whole server to gather together. I think that the person who found the mark has every right to do whatever they wish with it. If their group can defeat it on their own, because they took the time to hunt it or were just lucky and found it, then so be it. If they want to share the hunt and wait, more power to them. I just don\\'t think actual legitimate hunting parties should have any obligation to wait on the rest of Eorzea.
Imposing some sort of immunity "timer" when they spawn as you suggested would just be detrimental to those who actually hunt the hunts, forcing them to wait when it was never a requirement to wait.
If you really want to be serious about getting Allied Seals, join a hunt party and actually hunt. Otherwise, accept your luck in the meantime.