The Web: Does Page Performance Have Any Correlation with Search Engine Optimization?

Big NO and small yes.

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No, first
Why not?
Let us read an example. Yes, let's.

On those sweet testing places (Google PageSpeed, Pingdom, WebPageTest, GTMetrix, and all), we can test a blank HTML page, with proper "viewport" and "charset" meta tags, plus compressing the transfer in gzip, we'll get nearly/perfectly perfect score. Like, all of the criteria: user experience, network resources performance, and the other stuffs.

But who will go to that blank page?
There's nothing on it. Well, there's that solid color...
Except, if you spam all your friends/relatives and/or your social medias followers/friends to visit that URL. And somehow, because you were adequately funny, they will land safely on that page.
Then, they'll get confused. Furthermore, if they're angry enough, they'll remove and/or block you from their "following"/"friend" list.

Is that winning or not, I can't tell.

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Second, yes. Small
If you've already had like, more than couple interesting articles and/or online service on that domain, then at some point think, "boy, I wanna speed up the page load, and redesign or sumthin", and miraculously succeed, some of the returning visitors will be like, "whoa, neat. I've got to recommend this URL to more living beings".

Now that's the "yes" part.
But that will rarely happen. Mostly like, meh. Nerd.

Anywho, even though rare, that URL will have more incoming/inbound links. Which will be detected/updated by crawlers as an important search data figure.

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In conclusion
The Search Engine Optimization (SEO) is still about the content of the page, the relevancy of the search keywords, the way we present the thing we show, the inbound recommendations, and more more such.
It's still a very obscure thing to learn, like random, but, in time, we'll "understand". Or not. Who cares?
Well, I do.

Then mwaaaayy at the bottom, is the page performance/user experience.
You'd notice some of big famous websites have weird slow/non-responsive thing. But, they're still big and famous.
Because they do provide things we can't resist.
Haha?
Indeed.

So, the order of getting traffic on the web would be like so:

1. Your sweet content/idea/service
2. Create a decent page layout
3. ..SEO..
4. ..advertising maybe..
5. ..analyzing the analytics perhaps..
6. Revamp the sweet content/ideas/service
7. ..es ee ow...
35. ..another ad, maybe..
57. ..cool link exchange, might..
69. ..sumthin..
101. Optimize that page, if you care enough