Air carrier Captain reported low altitude alert on approach. Flight crew corrected and continued approach.

Narrative

On flight from ZZZ1 to ZZZ; I was pilot monitoring and the First Officer was pilot flying. The flight went well until the approach. We flew the ZZZZZ arrival with transition to XXL. The weather was perfect. We had flaps set to 20 and ZZZ gave us a left turn for a base. He finished configuring; and I recall telling him to be careful with his airspeed; as he slowed down; he set REF+5 a bit early. ATC asked if we had the airport in sight and we did. They cleared us to turn left to a certain heading and cleared us for the visual approach. We were at 3000ft and close to ZZZZZ1. I saw the glide slope go by and we were too high. Things went extremely fast and a lot was going on; I cannot remember very much of what happened in what order. But because he was high; I told the First Officer I recommend you hand-fly; we are high"; so he did. He disconnected the autopilot and flew manually. He might have been in a bit of a turn from the final intercept heading; I am not sure. To avoid him descending too fast; I gave him VS 1000 fpm since we were inside the FAF. Between that and switching to tower/running landing checklist; I was heads down. I looked up and all of a sudden see that he's about to line up with XXR instead of XXL. I told him that; and he was a bit slow to correct; so I momentarily grabbed the yoke and encouraged the left turn. He corrected. Between focusing on the runway alignment outside and still trying to run the checklist; check in with tower; I did not notice he was getting low. We got the "glide slope" aural warning and he corrected. Then ATC also gave us a "low altitude alert" a little bit later; as we were correcting altitude. This all happened very quickly; because I remember being back on glide slope and all being normal between 700 and 500ft AGL. It is beyond me why I did not call for a go-around. It all went so fast; so much happened in such a short amount of time; I don't remember at what altitudes things happened. Our day started really poorly; with a late inbound and isle chair waiting to deplane. We had to move quickly; I tried to slow things down; not to feel rushed. But quickly; I felt pressure to move faster after push; for an inbound mainline that we were blocking. I think it kind of set the tone for the rest of the turn. On another note; I keep getting switched from late nights to early mornings back and forth; and having started very early; my focus was probably not the best. I should have been paying more attention during the localizer capture moment; and when things went wrong; I should have either called for a go-around or taken the controls. We debriefed it once at the gate. I should have asked for a go around. I think I got caught in the moment. I normally have no issues calling for one or executing one. I could have also taken control of the aircraft and done the go-around myself; when I noticed the wrong runway alignment or the glide slope."