Kayak, for instance, now sells all flights through third parties, like ChatDeal. The internet abounds with tales of ChatDeal not actually purchasing the tickets, or canceling them at the last minute. Any issues go through these third-party sites, who do not care about customer service or getting you to your destination.

The idea of buying any ticket not directly from the airline terrifies me now.

They actually do on some airlines. For instance, check out the first booking option here: https://www.google.com/travel/flights/booking?tfs=CBwQAhpHag...

That looks like it'll still result in a direct Spirit booking, though.

Yeah. It feels like they’re acting more like a travel agent. Kind of like what this original post would be doing.

It's unclear to me from the Duffel page which scenario happens. https://duffel.com/flights/airlines seems to indicate some airlines have a direct connection, but the vast majority go through something called Travelport.

If I ask the airline to cancel my booking, what happens? Do they do so? Does the Duffel markup come back to me?