If the OP's husband wanted his wife and kids involved in his sporting jaunts, he would already be doing this. Suggesting he take the kids, or that he involves jis family, would be a good way of provoking a confrontation - but the idea that he'd go 'oh, yeah, okay, great' is just not in the real world.
She says the team often car share to away games. He'd have to change that - or she'd have to drive off down the motorway after him, with the kids in tow, and then realistically, how much responsibility would he take for them?
If you have a partner who is happy to make their extra curricular activities a family thing, then great. Weekends away together, the kids all being part of it - lovely.
But there are plenty of people who see sport or other activities as time away from their children. How many golf dads take their young children with them? Seriously?
btw, I run, but fend off the suggestions that I take the buggy with me and run pushing it. I hate that, it limits me and I find it really uncomfortable, not to mention not being able to afford the super run and push buggies. So I go running without any small children with me.
But I go FOR AN HOUR when it has been agreed with my partner that it is convenient. Would I like to train more, and travel away to races like I used to. Yeah. But a few years not doing that won't kill me. It would be hugely disruptive to family life if I insisted on doing that, regardless of what everyone else is doing.