I use a trailer for cargo. I.E. half the time empty, half the time full. There is no problem with it empty, other than being EASIER to pull.

You might ask your daycare facility if there is a place to park the trailer during the day, as you'll need it when you return to pick up the kiddo.

Most trailers are pretty easy to connect and disconnect.

Thule Chariot trailers look very nice, but are expensive. Perhaps start hunting for a good deal on a used one?

People with cargo bikes are happy with them for kids, and child seats might be good for the smallest kids. It is up to the parents, I've seen them all in use.