The drives are controlled by the CPU on the main controller board. The stepper windings are driven by the power amplifiers according to I/O line signals from parallel interface interface chips (6821 / 6520 PIO) in turn according to a sequential bit pattern from the CPU - whatever mode the machine is in.
In 'Hand' mode the CPU detects the +/- key taps and advances the pattern. When the +/- is held beyond a certain time the CPU uses pulses from a 555 (or equivalent) timer IC via another PIO. The 555 pulse rate is determine by an R/C (resistor/capacitor) circuit containing the jog speed potentiometer.
From what you say I understand the machine moves correctly in 'CNC' mode and also jogs one step per tap of the +/-.
That implies to me the CPU is reading the +/- buttons correctly and the drive circuit is good but that the pulse train is not correct.
My first test would be that the pot is in good order and that the connections to the board are ok. Pot spindles (knobs) can get bashed by mishandling leading to poor internal connections and/or the pot can come loose on the board which leads to poor/broken connections.
Less likely, any metal chips which may have found their way in to the main board past the shielding can also cause problems.