Must wrestle the stricken aircraft into the sky?
Not any more. This is a hang-over from decades ago in aviation when aircraft had to be hand flown in critical phases because automatics were; a) in their infancy, b) not very good, and, c) not very reliable. In addition, controls were not all hydraulically powered and were manually trimmed, so a large degree of manual flying skill, (and dare one say 'macho' attitude) was necessary.
Nowadays, thanks to years of research and development, modern automatics are extremely good and reliable. Even when manually flown, the airframes are much easier to fly, and an EFATO on an Airbus simply requires; Keep it straight with rudder, (then hold the rudder there); Rotate to +10 degrees, (then leave the pitch alone); Trim the rudder; Engage autopilot; Deal with the emergency; Land the aircraft.
If both autopilots are not available, obviously one pilot will have to hand fly, and this WILL take most of that pilot's capacity away from dealing with the emergency. But they will still be able to confirm correct guarded switch, master switch and fire push button selections etc., and deal with basic communications to ATC.
A point to be borne in mind though is that if one pilot is hand flying, then the workload in an emergency of the pilot-not-flying goes up massively - probably by 100%, because s/he is having to be much more vigilant in monitoring the PF's hand flying, (for example bank angle), AS WELL as dealing with the ECAM, checklists, and landing distance calculations etc. etc. So while PF might
think they have plenty of capacity to hand fly, they must remember that in doing so they are actually using some of PNF's capacity as well.
So PF might think they're Jack-the-lad and the poor dumb PNF is slow and behind the aircraft, without realising how much they have loaded up the PNF. They would do well to think very carefully before hand flying in an emergency unless there was no alternative.
U