To the customer it may be Delta. But it in fact is NOT Delta Airlines that flies the jet. Look at the side of the jet near the cockpit. That usually shows the regional airline that flies the plane. And look at the ticket. It will say (usually in small print) Operated By so-and-so Airline. The airlines flying the regional jets are completely separate airlines (except for American Eagle, which is owned by American Airlines). The only thing they have in common is that their tickets can be purchased from the Delta website, and the paint scheme looks the same. The regionals just bid to receive these routes that Delta (or any other major airline) doesn't want to fly, or can't fly, sometimes due to size limitations (i.e. jet is too big to land at a certain airport). Other times there is no market for major airlines and the feeder airlines (regionals) fly to these destinations.
If people are shocked by this, they obviously haven't read their ticked correctly. This has been happening for many, many years. Filing a complaint with Delta Airlines about something that happened on a regional jet with a regional airline will get you nowhere. The regional airlines are not franchises. They are literally their own airline.
I invite you to check out
www.raa.org. It is the Regional Airline Association and may help you better understand the role of regional airlines.