I wouldn't worry too much about the "running, jumping, climbing trees" thing and the "other kids' dads" business.
The other kids will be green with envy.
Your kid gets to go shooting on weekends. Gets to go fishing. Gets to learn all kinds of cool stuff that the other dads don't have time to teach.
I have a couple of granddaughters, aged 3 and 5 years. Their dad and mom (my daughter) have the typical frantic working-for-a-living life. My daughter has somehow (hey, don't look at me) learned that, regardless of what else is going on, it's important to spend weekend time with the girls. However, they can't afford some of the toys I have, so when it's time for the range, it will be grampa's guns and ammo they learn with.
I've lived a little bit of life that not everyone gets to, so I have some other things I can teach. And I have other things I will want to teach, but I must first learn them myself (or learn again, in some cases).
By the time the older girl is ten, I expect her to be able to tie every knot in the book, start a fire without matches, and make shelter. Oh, and knock over ten consecutive tin cans at 30 yards.
I don't run and jump much any more, but I can still toss a mean Frisbee.
Play to your strengths.
Make one of those strengths "time and understanding."
You'll be fine.