I would do it if it was in my area and some of the other girls fancied it and it didn't clash with anything. There's a relay that my club puts on called Devil's Burdens that does attract some pretty decent runners (winning team typically all sub 16 5k runners) but also all kinds of other runners too. Though probably not so much people who just do parkrun as it's in the hills, in winter, and some of the legs have to be run in pairs.
Actually, they can subsitute for each other too. E.g. if you want to get a good time, you might enter a race with a high-quality field where you won't be anywhere near the front but will get dragged along to a PB. Or if it's a random distance or poor conditions you might not be expecting a PB but enter a more low-key race and try and win it. Or alternatively, you might enter hoping to win or place but someone much faster turns up, in which case you might think 'well, I didn't win but at least I got a good time', or vice versa if you run a crap time but still win something. But mostly I agree with this:
When I joined there were only 3 female members (I was the 4th). They have more now, but you need a LOT more than 4 to guarantee having a team for a competition. A lot of races and championships have male teams of four but female teams of three because there just isn't the female participation. And it was pretty rare for us to have a full team for the county champs or xc even when more women joined.
I think Dean and I are looking at it more from the perspective of people who go to events of all kinds and know race organisers etc than as people who would want to turn up and win.
Yep, 5k legs definitely more fun as if you have less experienced runners it might take them longer to run 10k which makes it into more of a chore than a fun team event. You'd want to advertise to local clubs and also at parkruns, and you'd have to have male, female and mixed team categories to appeal to more people - even in my old club we'd have struggled to get a team of four women out. Also I think a 'family' fun day based on running with no kids' race would be a bit disappointing.
Re price, I can imagine some of my teammates doing this if it were local to us and there were decent prizes, but we don't have kids, so we wouldn't be into paying loads to fund the activities. Maybe you'd want to look into making money from refreshment stalls etc so that runners wouldn't feel like they were paying for other people's kids.
One time I twisted my ankle, felt fine a week later, resumed running and... then after 10 days of that developed foot pain and had to take two months off. Hopefully that won't happen to you but I would agree with the others that there is more to be lost by trying to catch up with missed training; prioritising proper recovery is more important, even if you think you are okay.