You are correct.
Anzuru yori umu ga yasushi (not yasashi but yasushi because it is old Japanese)
is exact what you said.
Anzuru yori umu ga yasushi is one of Japanese old proverb.
Not only about delivery but more in general, its meaning is "Don't worry about that."
or "You don't have to worry about that."
A pregnant woman tends to think the pain, suffer, safety, financial problems etc. about delivery beforehand. But after she has finished the delivery, she says that it is easier than she had imagined. So this proverb was invented by old people.
You can use this proverb like this;
When your friend have to take an operation in his stomach and he worries a lot, you can encourage him by saying; anzuru yori umu ga yasushi desuyo.