TRACON Supervisor reported a NMAC had occurred between VFR arrival and departing traffic from a satellite airport.

Narrative

I was working the desk and I had split. I had two strong experienced Controllers working; I was attending to the facility rotation when I heard the Conflict Alert going off. I got up and walked to the radar scope and asked what had happened; and I was given a complaint about ZZZ turning an arrival aircraft that was VFR and the alert went off with an IFR departure that we were talking to. But I caught the tail end of the situation and the controllers didn't make the situation seem like something that I needed to follow up on. I just gave an on-the-spot correction on how I thought that feeding the Tower arrivals straight at the airport without using down winds is not good practice and it ties both the Tower and radars hands. I also thought it all happened in Towers class D airspace; making them responsible for arrival/departure situation. The next day; the controller mentioned that he wanted to talk to management about the ZZZ contract Tower and complain about their working habits. I then pulled up the replay and saw how ugly the situation was. It turns out that Aircraft X was doing a practice approach with climbout instructions and the controller gave the Tower two VFR arrivals aimed right at the departure corridor; and when Aircraft X was on the go; there was a near mid-air between Aircraft X and an inbound Aircraft Y. In the future I need to not be complacent in slow periods of traffic. Next time I hear controllers complaining about traffic situation; follow up and pull up a replay on it.