[old_tv_nut]

I think the DC coupling from C2 to the AGC/sync circuits is crucial.
I think normal operation is that AGC sets the (negative ) sync peaks to the desired level, thus setting the overall amplitude and the DC level of the half point of the sync pulses for sync slicing. It looks like the two sync/AGC tube sections are DC coupled, meaning that they depend on the signal going negative from ground to put the sync slicing point at the correct level.

I think what might work is to build a DC restorer/amplifier circuit for the input that puts the bottom of sync tips at the correct negative voltage at C2 (about minus 4.5 volts?) while amplifying the signal to 4 V p-p. You will need a negative supply for this.