He said I could just leave it the way it is - with a small
hole where the water came in. I can feel no cold air coming out
anywhere and it appears to be working fine.

Oh no...you should definitely plug it no matter how small. Even if you
only put some tape over the inside and another piece over the outside. If
there is a thickness being spanned then maybe stuff something in there like
a wisp of fiberglass, a sliver of a Styrofoam packing peanut or even some
plumber's putty or caulk and then tape over that. If it's just a rubber
grommet with a 1/4" hole then just tape both inside and out. Electricity
is too expensive to leave any hole.

The freezer is still a
lot colder on it than on the new one. Guess it will catch up.

A new fridge can take a surprisingly long time to cool down. The machinery
can fill the compartments with cold air in minutes but there's a lot of
thermal mass with the inner walls, racks, etc. that take time to chill
down.