I actually don't mind the occasional ridiculous coincidence, in fact sometimes I'll play it up for the amusement.
Sometimes we'll do more lighthearted campaigns. They're short though. I like a good 6-8 session space opera for that sort of thing. Not in my regular serious game though.
Absent any qualifiers this is a bit of a red flag, in that what if your players decide, in character during play, to take things completely off-story? Would you let it happen?
Not usually no. When i start a new campaign I tell the group what the theme and general shape is going to be and usually I dont want to get too far off of that. Not every session is story either though. There are pauses where theres nothing really related to the overall plot going on and the PC's have a little breathing room to try to take control of their own destinies again. Kind of like how a season of a tv show will have a general theme and a climax in mind for the finale but not every episode involves that plot. Some of them are just random monster of the week type stuff.
There have been exceptions where they came up with an idea that seemed really cool at the moment and I went with it. But thats rare. An example I can think of was my last changeling game. First time any of us had played that one and the theme is generally kidnap victims of the fey find their way back to the mortal world and deal with their new altered life circumstances.
Most of the time you start after the characters return to the real world. But I decided for this one it would be fun to have them meet up in fairy and escape together. Unfortunately they threw a wrench in the works by deciding to cut a deal with their fairy master to go be hunters for the fey in the mortal world capturing slaves and working their will. Here I'm thinking they plan to break the deal and I've just been handed a BBEG. Nope. They honored the deal and went through a 9 month campaign as evil fey kidnappers and spies manipulating mortals into being part of the feys insane plans.
Not at all what I had in mind and I had to improv a ton in every session. But it was a group mainly of friends that I had played with for years and the new people seemed like good players so I trusted their direction and went with it. It was a fun campaign until they all got killed by the men in black, but I still kinda miss the original idea I had.
That was a rare exception though because I knew the other players well and we werent deep into the campaign. They put it off the rails right in the first session and when i told them that meant I was going to have to make up a lot of stuff as we went they were okay with that.
The challenge seemed kind of fun for me. Its not something I want to do every time though.
I'm somewhere in between.
I don't ask that players develop backgrounds for their characters but if one does so anyway I'd like to have at least a vague idea of what's in it; not so I can bend the game toward it but so I can incorporate it if the game otherwise happens to run on to it:
<party has to travel through a somewhat-dangerous mountain pass en route to their next adventure>
"Aloysius, you did time in the 7th Legion before you took up adventuring, didn't you? They were stationed in this pass for a few seasons during that time, meaning you know the area at least a bit and probably have a few local contacts in the villages. The 9th hold the pass these days..."
I want your elevator pitch. Sort of a cross between what you would put in your intro paragraph on a dating site and a job interview.
" I'm 37, divorced, have some kids, did a few years in the army and then was a traveling sales consultant for 15 years. Grew up a little rough but got straightened out in the service and now I'm a socially conservative Asatru pagan whose a recovering workaholic and whose kids are old enough not to want to hang out with dad on the weekends so I'm back on the road".
Tells a GM all sorts of things about how they can expect my character to react in various circumstances, and why I'm out searching for adventure. If they wanted to use me for various local info like you said then sure, i traveled around a lot and talked to people for a living. I'll play exposition tool for your narrator if you want. But I'm not filling out who all these people were or where i went ahead of time. And if someone did I wouldnt feel obligated in the slightest to use those details as a GM. Or to reshape one of my towns compositions because you wrote down that your buddy buddy with the mayor of said town.
If I were a player I wouldnt appreciate the GM dragging that characters ex-wife or children into an adventure as some sort of hook. So I wont do it to them. I make my characters backstories short and relatively devoid of details but whats there is mine and i dont want it altered or screwed with. So again, I dont do something to a player that I wouldnt want done to me if the roles were reversed.