I've got active children despite DS1 being autistic, dyspraxic and hypermobile, but it is an advantage that they've grown up seeing me being active and that it's a normal way to live to them. I'm one of the core volunteers with our junior parkrun so I'm either helping out, or sometimes when I'm free, run it "with" them. I say "with"... they are faster... by several minutes, but that's part of the fun for them. Sometimes it's not their day and that's fine. I ask them to participate only, if they're keeping the tail walker company, that's fine. Some runs just suck, and some days you just don't have the mojo and that's fine and a useful thing to learn and accept.
You don't need much to eat before parkrun, a drink of water and something light like a banana is great and will help prevent a stitch. Drink well the night before too as dehydration can be a trigger for stitch.
Getting into 5k is worth doing, but you do need the stamina to walk it for towards an hour, or energy to run/ walk faster. It's more appealing when the milder weather comes in the spring than now when it alternates between wet/ windy or dry and cold.
My two also do karate. It's sociable but without the team pressure that can put some off sport (which definitely includes DS1)
Then we have swimming. Lessons and some recreational splashing around sessions. I've always mucked in and played with them which helps; plus it keeps me warm compared to bobbing at the edge!
If you can do things little and often, that helps. Our school run is short, but because DS2 is faffy, we normally have to do it quite briskly which is good. DS1 needs driving now, but I park further away and it's further to my car than it was from primary to home!
Fitness is a different issue to weight (although excess weight can inhibit fitness). It's frustrating being slow and tired, and puts you off other opportunities to try things. There are so many fun things to try and fitness brings confidence and social opportunities. We've got some Cubs that struggle with activities like climbing walls and assault courses due to lack of strength and they are incredibly slow on walks and find it hard going; that's not an easy, enjoyable life for them.
It is worth reviewing diet and making sure that it is nutritious and matches a child's needs. I'm tweaking ours at present as we've slipped into sloppy habits about fresh cooking and snacking.