Separate tickets and interlining are not the same issue.

An interline agreement with CM and DL covers when one ticket has a flight on each of the airlines. Baggage interlining says for one common ticket they will handle the luggage across airlines. Ticket interline agreements say one airline can sell the others' tickets. You are not discussing an interline agreement at all. Especially since there could easily be a codeshare CM/UA flight that probably met interline conditions. Or just a mix of CM and DL flights. So you did not "get an interline agreement" since the agreements are between airlines and require one ticket, and more than one airline. You got an exception to a rule.

Airlines know full well that people will buy separate tickets as you did, to save money over a single ticket (the e-ticket interline agreements are how you get one ticket on two airlines, even ones that are not in the same alliance). So the airlines are saying two tickets are your problem and not theirs.