The reason the search bar was not showing was that there were only computed columns present in the Custom Collection’s display. Once a basic column value was added and displayed in the Custom Collection, the search bar appeared right away.
Another issue came from this though — the search bar in Custom Collection does not seem to search any of the computed columns, only the basic ones. This appears to be a limitation of the Custom Collection, but I’ve asked the team to confirm this and will update this post once I hear back.