JPMiller, in contrast to most everything else written here: My main hunting buddies and I took our kids (four boys, three girls) on a lot of trips, hunting, hiking, shed hunting, scouting, cold creek swimming, frog finding, etc. and pushed them much harder than most, apparently. Lots of miles, carry some of their own stuff--more as age allowed, all day, make their own fires, carry the elk bones they want to take home, push their own bike when they were too tired to pedal, etc. Sure, we had lots of snacks, kept them hydrated, carried them on shoulders or in packs when they were 3 and too short to hike all day.
Those kids all developed different desires for the outdoors as adults based on their personalities, but all of them still enjoy the outdoors and find renewal in the wilderness. At least two get more outdoor time as adults than any of us parents ever did. I've backpacked with them as adults in the Hells Canyon, Bear Tooth, Flat Tops, Cloud Peak, and Pecos Wildernesses on 5-day trips. Headed to the Sawtooths with them soon. My buddies' kids are adamant that we get together at least once a year in the wilderness.
My parents did the same with me. Me and my 5 siblings all backpacked as teenagers carrying all of our own gear. My week-long, backpack hunting trips started at age 14 in the Wenaha Wilderness. Those times were important in developing my desire for the outdoors.
Kids are resilient and usually in far better physical shape than their mental toughness. Grit is learned from tough and demanding times. Demanding doesn't necessarily mean "fun," but "enjoyable," "worthwhile," and "rewarding" may be more valuable than "fun." I'll probably get bashed here for not being PC and for being old-school, so I'll guess I have to just rely on our unique and particular results.
Good for you for desiring to get your three-year-old out in the woods. I hope its the beginning of many trips.