I halfway agree with your second opinion. You rarely need to push a negative film because of the exposure latitude. With films available up to 1600 ISO this is rarely necessary.

However:

to push a color film is actually surprisingly similar to pushing a b/w film

In my career as photographer I've pushed both, positive and negative films whenever necessary. The results with the E6 process were better than the ones from the C41 process.

Process requirements

You need to chenge the time of the first development stage only. The rest of the processing steps should not be changed at all.

With a linear machine like the colenta 30 E6 26 PRO there little options to change the development time. You can only change the speed of the machine, but this affects all chemical steps the same way. You cannot reduce the development time below a certain margin (because the fixation step will not be finished then) and your maximum increase is also only around twice the time.

I have used a Colenta 130 machine (spinning drum type) which lets you set the first development time to anything between 0 and 30 minutes. The same is possible with other "processors" like the Jomo drum or with the manual processors or with the good-ole Ilford box

Paramount to all color processing is stable temperature. As this temperature (38 degrees Celsius) is significantly higher than with B/W processing this might be a challange.

Experience with Pushing Color films

It is possible to apply the experience from b/w films to color films:

• Pushing increases grain
• It also increases contrast
• the opposite (overexpose and lower development time) can be used to reduce contrast
• it is better to push already high sensitive films further than to push a lower sensitive film to a higher ISO. For example: it is better to push 1600 ISO to 6400 ISO than to push 100 ISO to 200 ISO.

Pushing of negative films (C41 process) usually causes them to loose calibration with the automated printing processes (these are usually calibrated to color film types available on the market). This means: if you want a decent print quality, you may have to order a manual print.

Pushing of positive films (E6 process) is straightforward. Up to two F-stops is usually not a big deal. Starting from three F-stops onward you will notice the "blacks" to brighten up. I would put the limit there. There is little color-shift(*) below two F-stops. The color-shift that becomes visible above that, is towards blue or green. But this stems mainly from the "brightening" of the blacks. The black becomes dark green.

Note: The E6 process additionally allows to control the yellow-blue balance chemically by setting the PH-value in the color-developer. This can be used to reduce a color-shift caused by the pushing.