I've noticed that while QMH and CMH lamps runs up very fast on electronic ballasts compared to magnetic ballasts, HPS lamps runs up the same speed on both electronic ballasts and magnetic ballasts.
Why electronic HID ballasts can't accelerate the run-up time of HPS lamps?

Isn't the warmup time of HID lamps largely dependant on the discharge current?

If I remember, constant wattage ballasts, by maintaining a fairly constant current, extend the warmup time by keeping the lamp current close to (or even under) its final value. HX and reactor types on the other hand allow a cold lamp and its low voltage drop to draw more current and thus warm up faster.

Electronic ballasts probably have an even tighter current regulation than any magnetic ballast. I guess it's possible to design an electronic ballast that purposefully induces a large overcurrent in the first minute or so to speed up the warm up, then bring the current back to nominal value. In that case there probably are practical reasons to design MH ballasts this way which is largely irrelevant with HPS ballasts. In indoor applications I assume it's desirable to have HID lamps that get to full brightness as fast as possible. As HPS lamps are used outdoors for the most part, the time they take to get to full brightness isn't as critical.

Electronic ballasts can tightly regulate whatever parameter their designer wanted.
It is true the most inner regulation loop uses to control the arc current. But maybe except fluorescents, the target current is never a constant, but rather is actively steered to reach desired lamp performance.
With MH it means stabilizing the pressure the arc operates in, the intention being to maintain the correct ratio of the various luminous components in the fill, so its color. This yields to the use of higher current until the lamp is really warmed up, hence the fast startup.
With HPS (and now I'm guessing) the aim would be to maintain the light output, so keep the real power somewhat constant (this then inherently supresses the thermal feedback). That means the warmup follows rather normal settling.
Either way the ballast never uses any current above the startup current rating (usually 1.6x nominal arc current), so to not overload the electrodes.

The run-up time would also be determined by things like how many hours the lamp has done and how cold the lamp is before it starts.