In the case of e.g. cereal crops in the north, there are 4-6 months where you can do little beyond some maintenance, buying/selling equipment, planning next year's crop, etc. that means there is plenty of time for vacations. You are right that during seeding or harvest (especially), all bets for travel are off.
But, it is incorrect to imply that because you farm, you ought to kiss vacations good-bye.
Chickens are likely easiest, but on a scale to produce a living for a family, I don't think any type of farming or ranching is low effort or easy to step away from.
That seems to be what the article is about. The first case study is a vegetable farm, without any hint that they produce anything else. They are also industrial-scale, else they would be considered gardeners, not farmers.
Hell, I don't even have any pets because I travel frequently enough that they'd be a hassle. I can't imagine how I'd deal with an entire farm in these circumstances. I barely keep one houseplant alive as it is, not from lack of care when I'm home, but from lack of care when I'm gone.