EMB-175 Captain reported a loss of aircraft control while new First Officer was flying a manual approach. Captain regained control and landed after executing a go around.

Narrative

After briefing and expecting a visual approach to XXR into ZZZ; ATC advised us to expect XYR with relatively short notice on ZZZZZ arrival and cleared us direct to the ZZZZZ IF to intercept the localizer. Things got busy as we quickly briefed the changes and had the runway in sight and were soon cleared for the visual approach. At or shortly after crossing ZZZZZ (I believe we were at or descending to 4000'); I noticed that we had passed the glideslope; which was now below us. Neither myself; the First Officer (FO); or the jump-seater; who is typed in the E175; could identify why the GS hadn't captured; but the approach was definitely armed (my best guess is the LOC hadn't captured when we passed the GS; but the needle was green/captured by the time I recognized the situation; having been distracted with the runway change; radios; etc.) Whatever the cause; after a brief effort to capture the GS from above; I determined that we were too close and too high and needed to break off and try again; so we leveled at 3000 FT and ATC provided vectors for the same approach.We were busy all the way back around to final; but it was essentially a routine approach until the FO disengaged the Autopilot at or just below 500'. From there; things degenerated quickly. She was a couple weeks off IOE with (much?) less than 100 hours and had mentioned during the departure briefing in ZZZ1 that she found it difficult to align the aircraft on the centerline; which I understood to pertain somehow to takeoffs and wasn't thinking about at this point. But after disconnecting the Autopilot; she indeed drifted into the wind; right of centerline. She made a correction to stop the drift but was still too far right; at which point I called out the deviation; but really should have called for a go-around. She banked further left and it appeared that she would be able to land with the centerline within the main wheels; if slightly off center; but then did not correct the track; which remained about 10 degrees to the left as we crossed fully over the centerline around 5-10' AGL. I judged that if we landed in that state; we would likely collapse the gear; strike the wingtip; and/or continue our trajectory off the runway; so I took the controls and went around. Normally; I would avoid taking the controls for a go-around at such a critical point; but she didn't have proper control of the flightpath; and I felt that a crash was imminent had I not done so. Clearly; I should have called for the go-around well before reaching that critical point.By the time we leveled off; we had reached our minimum fuel reserve and advised ATC accordingly. They changed our runway assignment to XXR and vectored us around with some priority to land without further incident.It's worth noting that; following my biological wake-up time of XA:15 and the massive adrenaline spike as everything went from zero to 100 in a single heartbeat; things were pretty surreal between the go-around call and control exchange until touchdown. It took considerable effort to recover from the undesired aircraft state and restore a functional level of flight crew coordination and situational awareness as I was task-saturated and striving to keep my frontal lobe engaged (the first radio call I remember comprehending was someone saying; Yeah; that was pretty close.") The jump seater was an especially significant asset for the first minute or so in helping the FO get back in the game in the role of pilot monitoring (PM) while I was sorting myself out and flying the airplane.I can't overstate the impact of my fatigue throughout the event. The ZZZ1 turn was added to my schedule when I returned to ZZZ around XE:30. I knew I was tired but didn't feel it warranted a fatigue call--nor would it have; perhaps; under typical circumstances. But this was a high-energy event; and I was definitely in a low energy state when it began around XI:20. Not only had I been awake and working for eight hours at that point; butthe XA:15 wake up time was deep in the middle of my normal sleep cycle. I made what effort I could to adjust my sleep pattern for the early morning but had only received the assignment two days prior. My cognitive and physiological performance was slow and strained; although I didn't perceive it until the action had already started.Also; while it's not my desire or intention to disparage the FO; she was very new and; as a pilot; not as strong as the other new-hire FOs I've flown with in recent weeks. But this wasn't immediately noticeable until Autopilot disengagement either; this being the first time I'd flown with her as pilot flying (PF). I believe my fatigue; coupled with a sense of complacency based on the performance of other new FOs; accounts for the delay in my recognition and appropriate response to the seriousness of the situation. I allowed it to go too far--within a split second of disaster; I believe.Suggestions: I've studied; taught; and applied key concepts of aviation human performance throughout my career; but understanding the effects of certain scenarios and experiencing them are two very different things. The cognitive and physiological disruptions I experienced during this event were real; involuntary; and frankly harrowing responses. And I'm now more convinced than ever that the only effective countermeasure is deliberate; conscientious adherence to SOP. I should have called for the go-around as soon as the flight deviated from the stable approach criteria; period.As for the PF; I think a bit more (remedial and non-punitive) time and help to sort out her landings would be beneficial. If I'd had more time; I would have worked with her myself; but another unfamiliar Captain might be caught off-guard as I was and; again; I believe any further delay or lack of response would probably have been disastrous.Finally; it's my understanding that we use the scheduling app or similar software when building scheduled flight crew sequences; lines etc. But I don't think we're using it to evaluate or predict fatigue in daily scheduling operations for reserves; added flight segments; reassignments and other modifications; etc. If that's correct; or close; it needs to be addressed immediately. ALL flight assignments should be run through such a program and confirmed to adhere to a standardized threshold of fatigue risk. There is no justification; even financially (81 souls plus the jet = not worth it); to do otherwise. I would be curious to know what my "blood alcohol equivalent" rating would have been at the time of this event."