It honestly depends, for me, on how long it takes for mods to update. If I was using a lot of key mods or CC that broke with a patch and I don't know when/if they'll update... odds are, I won't return to the save even once the mods/cc are updated. "Key" in this case being mods that actively change gameplay, such as the chemistry system mod, or else custom traits/careers/aspirations/etc.
It's actually one of the reasons why I've been trying to cull my mods so much, and why I try to run test saves before replacing my mods after I get a new pack - to remind myself that the game doesn't require mods for me to have fun with it.
But I do get what you're saying. There's been more than one pack/patch that has made me want to emulate GalacticGal and start over, because it changed something key to my gameplay in that particular story... usually for the better, or it would have been if I'd had it before.
As for what I do about it... I usually just start a new story. I see nothing wrong with moving on when the story no longer works for you. Now, if this means you're dropping your story every month... I can see where that might get concerning. But if the pack doesn't offer any direct impact on your current story (like Werewolves for most Realism players) then just try to see it as new families moving in to those shell houses. ;)
