I could easily agree with it becoming central for a Fox main. His PP seems to be a cut above the rest. Even Izaw saw it in his original video of zaw dancing. Everyday I use it more and start to wish I could implement it more consistently in various situations. His PP makes it much easier to micromanage spacing and stay "in" with the opponent without sacrificing options. However, i do think stutter stepping should be just as integral as PP, which I recall you stated as being key as well.
I think that running will still have a use though from time to time. Particularly against other speedy characters. PP shield lags a bit on reaction times (at least for me) because I have a brief moment of helplessness from the initial dash of the PP. A slow character would have a hard time surprising me since it is only 1 frame, but sonic or another fox could very well adapt to the spacing of the PP and expect it at high level play if the person only moves using PP. Stutter step does help this, but running in these cases could give more leniency in certain scenarios to react better to an attack. We'll see though as the meta develops whether or not this is true, but even with wavedashing, it did not become the sole movement option. It was integrated with all of the other movement options, and with time was implemented more, but never phased out the others entirely. Well, with exception to some characters like falco. I seldom see a falco just run in, he usually either wavedashes, or SH lasers the opponent into a grab as an approach, but there is good reason for that.
That is the only disagreement I kind of had, but I do agree that PP for fox at least, will become a core movement option interspersed with running, dashing, and stutter stepping. If PP becomes as big as you predict (and it might, but some top players are skeptical like JTails, and I do respect his opinions), then I could see the ratio of PP to dashing and stutter stepping leaning towards the PP. Jtails has stated that he thinks PP off of landing from a jump has great utility though.
Still, there will be some characters who benefit less from PP, and perhaps should seldom use it. Jiggly Puff from melee and Sm4sh are good examples of a char who does not NEED these options as much to compete against people who have mastered them. You will seldom see a good melee Puff wavedash with good reason, just as you probably will rarely see it PP. I think diddy and peach are two others that would benefit less from mastering PP with them, as it seems to be strongly tied to movement in the air, which both characters have much better horizontal movement than fox while in the air. Fox is a very grounded character that uses short hops in conjunction to ground movement to pick off aerial opponents. So it is not surprising that we perceive his PP a being a nice option.
It is a shame, you are quite right that we probably could not play, at least without tons of nasty lagggg
I enjoy the mental exercises that you are making me do though, which is why I want to play you lol