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For a time. Once one of those 9 leaves, the 10th kid will get daycare. If, however, the 10th kid was behind a bunch of non-born/non-conceived kids, that 10th already-existing kid might not actually get daycare.
The problem is too few places.
Once there are enough places, early sign up is preferred. This allows them to plan effectively, e.g. by hiring and training enough people in advance. It also allows parents to plan ahead.
An institution fails to deliver a crucial service and you blame fellow victims. >

If no places allowed early (read: pre-birth/pre-conception) signup, then the problem of too few places would be at least partially alleviated because those who were born at the time their parents put them on the waitlist would actually get places within a reasonable timeframe.
I'm not saying eliminate waitlists entirely. I'm saying eliminate putting not-yet-existing kids on said waitlists.
Jumping to the front of a line before you have something to be in line for (a baby, or in some cases even a pregnancy) is what's mucking up the system and making fewer places for those kids who are already actually here. >

How?
There are 9 places in the daycare. There are 10 kids that need it. How will signing up later change anything about the missing place for 1 kid? >

The one kid will still have to be on the waitlist, but that kid will be at the top of the list because there aren't any kids that aren't even born (or possibly conceived) yet on the list. So that one kid will have to wait less time than would have to be waited if people put a kid on the waitlist as soon as it was conceived (or even before!). >

In both scenarios only 9 out of 10 kids will get daycare, right?