I think it often depends on two things: the child himself and the area you hunt.
I began taking my two daughters, who are now in college, hunting here in NE Oklahoma when they were about 10- and 8-years old.
Unfortunately, Oklahoma didn't allow feeders at the time and the ranch was primarily open pasture, with woods in the creek bottoms and on the steep hills. I built stands along deer trails and in the bottom land surrounding the creeks. As a result, sometimes I'd see several deer during a morning or evening hunt, but many times I wouldn't.
The first few times I took them with me, we were unlucky and didn't see many deer. After sitting in the stand for a couple of hours watching only squirrels and birds, the kids would start to get antsy, particularly on the morning hunts, after getting up so early.
If you're hunting in Texas, where most counties have prodigious numbers of deer and turkey, and hunt near feeders or food plots, you probably won't get "skunked" and not see any deer. The youngsters won't be as apt to lose their patience, if they've got something to watch.
Also, my friend takes his 6-year old son on evening hunts only. He finds that his son can handle those better, since he doesn't have to get up so early.
One danger in taking kids when they're too young is they'll get bored quickly and decide that it's not fun. That may make them less likely to want to go when they're older.
Fortunately for me, I now hunt with my daughters at my friend's ranch in Comanche County, TX, where there are plent of deer and turkey. They are now avid hunters and won't miss a chance to go.