How busy is too busy?
I'd think we're pretty far right on the busyness spectrum but we have no problems with keeping chickens. On weekdays we usually get home at 9 or even 10pm, and we leave around 8. We feed our chickens once a day, in the morning, and they have a half of the garden to dig up insects and find seeds in. They lay large eggs every day and our friends always comment on how they've never seen fatter, more healthy-looking chickens.
As we only see them (the chickens, not the friends, lol) for maximum five minutes most days, I think they are excellent at taking care of themselves.
On weekends we let them out to the front half of the garden while we keep an eye on them, so they can dig beneath the big trees. We save food scraps daily to give the chickens after their morning meal. We've never had a problem with foxes (there are no foxes in this part of the city) or other predators, the same as other households with chooks in our suburb, and our chickens have never been sick besides the usual symptoms of old age. (Besides once when chicken #2 ingested something unsavoury in the garden when she was 2 weeks old, but I don't count that.
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I think if people know the essentials for taking care of chickens and take action as soon as something goes wrong, they can successfully raise chickens even if they lead busy lives.