have you examined your HTTP Request and Response headers to see if fresh and cached requests respond as expected?

Can you elaborate more about where and what to check exactly?
I have been checking my Cloudflare's stats through Siteground>Site tools about the Uncached/Cached requests of the last 24 hours (for the last 3 days) and also through Redbot.org and Chrome's dev tools about the CF-Cache-Status. Unfortunately the MISS status is there for homepage and main subpages, when I guess after all these daily visits and with the use of the Cache Everything page rule they should have been cached by now.

see the documentation about how CF uses the Cache-Control and Expires headers:
[support.cloudflare.com...]

the value of these headers in HTTP Requests should help explain why you got a MISS in your CF-Cache-Status Response header.

The link that you have provided explains what do the different descriptions like HIT, MISS etc mean and says nothing about the Cache Everything page rule.

says nothing about the Cache Everything page rule

you apparently focused all of your attention on the "Understand Cloudflare cache responses" section.
i would encourage you to familiarize yourself with the Understand Cloudflare's default cache behavior section.

regarding Cache Everything:

The Cache Everything setting forces Cloudflare to consider all content that matches the page rule URL pattern as potentially cacheable. Whether we actually store the asset and for how long is determined by the Cache-Control headers returned with responses from the origin server. Learn more about Origin Cache Control [support.cloudflare.com].

(source: Page Rules limitations [support.cloudflare.com] )

a much more detailed description of CF caching here:
Understanding Origin Cache-Control [support.cloudflare.com]

Just a small update regarding the "Cache Everything page rule".
I have made tests with setting it on and off for some period of times.
Unfortunately my Uncached/Cached requests don't seem to change as I would expect, especially in favor of the the Cached requests. My Cached requests served are still at 27% for the last 24hours.
So except if some special technical setup is needed, it seems that it doesn't bring any noteworthy results to the average user.

what is the Browser Cache TTL setting [support.cloudflare.com]?

@phranque
Apologies, haven't seen your message earlier.

It's set at Respect Existing Headers.