Can the captain not have simply DISEGAGED JUST THE AUTO-THROTTLE and still kept the horizontal LNAV engaged (with autopilot still on, of course)?
I am sure he could, but would have required him to notice the ATHR was not working / the speed was low
I mean, will the ILS signal still guide the plane properly to the centre of the runway laterally if VNAV is disengaged?
I am not a 73 pilot, but woukd be very surprised if a 737 makes a coupled approach in LNAV / VNAV
I would think it would be like the Airbus in some sort of LOC / G/S modes...
Personally, if you are well down a glideslope, the ATHR "fails" (or seems to) and/or the speed get seriously low, your priority is probably not to figure out a way to continue the approach. I would suggest you throw it away / go-around, fly the aircraft first, then diagnose the problem and get both your SA back as to what is/was going on.... but as I said above, that relies on you noticing the "problem" in the first place.
NoD