D.B.
Welcome to Mamapedia. Despite the name, dads and grandparents are welcome here.
I think kids go through stages of only wanting to be with one parent or the other. It's very common. It sounds like Mommy is the one who does all the basic care, is in charge of schedules and dressing and meals and getting in the carseat and brushing teeth - all the things that kids balk at. She's also the one who's going to take him to the babysitter so being with Mom means losing Mom too. Dad is only there 60 minutes a day? That's so so so little. So the time is precious to your son and of course he is not going to do well when Dad leaves. Who can blame him?
I'm not sure it has to do with spoiling him when he was younger - that's usually in older kids. But if your wife is overburdened with all the childcare and tough decisions/separations/transitions, she may be a bit overwhelmed and tired of being the bad guy all the time. So she may be a little more short with your son. Can't say I blame her.
If you come home from work in the morning, is there any way that you can spend more time with your son after you wife leaves, and then you be the one to take him to the babysitter's? Are you going to bed exactly when you arrive home in the morning? If you have one car and your wife is taking it to work right after you arrive home in it, that's obviously a problem. But if you can spend more time with your son, it would be better for all 3 of you. If you are sleeping roughly 8 hours and working roughly 8 hours, then there are still 8 hours in the day. Maybe you spend 2 of those commuting. But there are still 6 hours left, and only 1 of those hours is going to your son (in 2 half-hour shifts). Maybe there's something you can work out with the remaining 5 hours??