I think that perfectly aligns with their take.

People in their 30s, married, tend to have more stable lives. They are in a position where they feel they are able to give that child a good life.

That actually makes sense. I think I broadly agree with this. Maybe we can do 100 different little things to help people feel like they are "set up".

Peak fertility though is early 20s. By the time people are in their 30s, having trouble getting pregnant is a more common problem.

Are people who want to have kids really struggling to have them though? Keep in mind that fertility rate is not actually about fertility (not directly at least). It's a measure of the kids being had rather than a measure of the capability that the name would lead people to think.

Ivf is way more frequent then it used to be

Exactly. So since we observe that actual birth rates among women in their 30s displays a different trend than that of women in their 20s, despite decreased natural fertility, there has to be more going on.

More and more women have the power to choose when they get pregnant every day.

This is the number one reason for the decrease in fertility. Unplanned pregnancies are becoming a thing of the past.