I have started all kinds of ways, but no matter what I try to come up with a premise that players are already aware of when they make characters and choose backgrounds.
In my two current Saltmarsh-derived games, the characters are all exiles from the "The Known World" (not 
that Known World - though it could be) that has room for any kind of background you want, with the understanding that that life is done, we're never going back there. I like to describe as kind of like Westeros or whatever - a place of convoluted history and back-stabbing and no such thing as a clean victory and nations are in constant war absorbing and splitting off from each other ad nauseum (also all orcs and goblinkind are wiped out, most non-human peoples are dwindling if present at all, and most monsters are extinct). Instead, thy come to some far corner of the world they have heard mention of, but that the "known world" thinks of as 1. to far away to practically conquer and 2. beneath its notice anyway). Here in this smaller micro-setting the PCs can do as they like, getting involved in politics or not, dungeon-delving, being general do-gooders, whatever but on a manageable scale. I had them meet on the long sea journey to where the campaign begins and thus we back-formulated what they know each other sing flashbacks to that journey and its talks.
The last campaign I ran "Second Son of a Second Son" had all the PCs be low-birth rank nobles of lower birth rank nobles (or bastards) who sign on to an adventuring charter to bring glory to their houses and hopefully raise their standing. Some were related to each other, some were from rival families, and some knew each other since childhood.
The campaign before that "Out of the Frying Pan" started with the premise that all the PCs were looking for a way to avoid conscription in the current war - thus insuring the main hook was taken (though being prepared for that not to happen). I have 
a complete story hour of this campaign that is readable in the original thread, but there are also compiled and cleaned up PDF downloads available in the first post.