It's a bit more nuanced.
Is child care as expensive as he says? Absolutely, it can easily be €600 per kid.
Thing is, if you make €20k, so €1600 household income, you get €520 or so per kid from the government for every €600 or so you spend. So you end up with €80 per month per kid, very doable for a single parent. No parent in the Netherlands makes €800 on full-time work, so we have to assume €1600 household income is for a single parent.
Even if your household income is €80k, i.e. 6-7k monthly income: typical for both parents working solid office jobs in their mid-late 30s, you still get a little over €200 per kid out of €600 expenses per kid, so €400 out of your more than €6k.
There are various caveats, and paying €80 for a single parent is no joke. But it's not nearly as bad as saying a median salary goes wholly to child care of two children, far from it. And it's unlikely to be the sole or even main reason for child care. In fact, I'd make a cautious bet that you'll see significant part-time statistics for people who have no children.
At the end of the day, the costs per working day per kid after subsidy is anywhere from €4 to €20. So the notion that this prevents people from going to work in order to be with their kid so as not to have to pay childcare, is false.
As on the low-end an individual makes like €13 per hour stacking shelves at a supermarket by age 25, working a 9 to 5 in a low-end job easily nets you €80, easy easy easy, at which point you make €1600 a month and pay the €4 a day rate. If you make say €40k and your spouse does, too, then you make more than €160 a day each, and so the notion that you'd rather stay home to save €20 and earn 0 than to pay €20 and earn €160 is unlikely. Even with two kids or three kids, it still pays off in a huge way to go to work.
Now there are some situations in which it could make sense. e.g. a parent who, somehow, has a 9 to 2 contract (5h a day), and picks up the kid from school. That can make a lot of sense in a family where the husband makes €60k and the wife could make €30k full-time, and €20k part-time in a 9 to 2 job. In this case the household income is such that it costs €20 per kid per day and it can make sense to work the $20k parttime job that allows you to pick your kids up from school (€80 income a working day) instead of the €30k parttime job (€120 income per working day), because if you have two kids, then childcare costs €40 per day, the difference in your working hours. And then it's probably more fun and useful to be with your kids than to pay for someone else to, if it's financially all the same. But 9 to 2 working days barely exist, and this is in the context of a cliched family where the man earns €60k and the woman has €30k max earning potential full-time. In more realistic scenarios the financial incentives are definitely such that it pays off for both parents to work full-time.