I haven't encountered anyone who had mistakenly thought that push processing would really increase the real film speed. Everyone understands it's a compromise.
But, I have encountered many who mistakenly think something absurd like "there is no push processing" or "push processing is a myth" or "push processing does nothing" etc. How hard it can be to understand? Push processing just means processing to higher contrast and it's done to push lower midtones and midtones higher, to get better Exposure Index; the downside is the compromised shadow detail. In addition, YES, it REALLY increases the real ISO speed also, --- but not much.
I remember reading KODAK document about ECN-2 negative pushing. It said that 1 stop push increases real speed by 1/3 stop and 2 stop push by 1/2 stop. That's about it. So, with 2 stop push you sacrifice 1 1/2 stops of shadow details but can get good lower midtones. The underexposure latitude in these films is often so good that decent results are still obtained. It still may have more latitude than digital cameras!
I mentioned pushing Vision3 500T to 1000 because I read somewhere (probably at Kodak's website...) an interview of some cinematographer who specially mentioned how well this new film handles pushing to 1000, even 2000. Edit: or it may have been 250D that he liked to push to 500 or 1000.
Movie industry needs pushing because there is a certain limit in exposure time because of the frame rate and shutter angle.
I've also pushed C-41 films with good results. It DOES help compared to underexpose+normal dev. Not as much as in BW world where even 3 stop pushes are common, but if you've underexposed, unintentionally or intentionally (not enough light to shoot), just push. It helps a little.
And, who does ECN-2 push, well, any movie lab. I haven't used labs though as I'm building my own.
If you want ECN-2 chemicals, you don't have to buy gigantic liquid concentrates. Just mix using raw chemicals. The formulas are in KODAK h2407 processing manual. It's easy, I've done it.