A digital display effectively operates the same as a CRT, compatibility will be a matter of the display's signal processing, just as it is with a CRT's analog signal processing circuits. In most cases digital displays are more flexible/forgiving because they aren't constrained to simple functions realizable with analog components.
Anyways, every display is calibrated to center images starting after a holdoff period once sync is triggered.
NTSC and PAL video line timing is close enough that the differences SHOULD be negligible, and all video should be approximately centered on any display, but some consoles/computers offset their active video slightly left or right of center, and also many TVs are calibrated off center too which exaggerates the incompatibility.
Arcade games actually do have different line timing because they aren't tied to broadcast standards.
What the cheap "shifter" device does in theory is delay sync approximately a line, adjustable with a potentiometer. They delay sync since it's much easier to delay a single pseudo-digital signal than it is to delay 3 analog video signals (would require a complex digital delay circuit). Unfortunately sync can't be perfectly delayed using a cheap RC integration circuits because the delay period is longer than the sync pulses themselves, so the "shifter" must create an artificial sync pulse triggered after the delay, which itself is triggered by the original sync. This artificial sync is completely malformed from the original sync since it can only do one kind of pulse. The original sync is modulated with special Vsync pulses too, and these will be distorted or even lost, but in most cases you can get away with it. Also as a consequence of delaying sync one line without delaying video, video is shifted upwards a line, but that's obviously a good tradeoff.
Analog "shifter" -- $3 of parts
A more flexible digital "shifter" -- $60 of parts + massive R&D, plus there's inherent signal degradation