Your post doesn't include the details necessary. What the Captain said about passengers at EWR isn't relevant to whether DL ought to pick up the tab for you at DTW.

Basic question is why was your flight late?

Your contract with DL lays out clearly what DL will and will not pay for. While it's short-hand, when the delay is within DL's control such as MX, DL will pay for a hotel. When it's beyond DL's control, WX, ATC and the like, DL won't and you are on your own.

Most travel insurance covers this stuff, so that's an easy starting point.