It is theory, but it isn't abstract. It's as straightforward as statistics gets without delving into the actual math. The explanations given are how RNG works. I couldn't care less about whiteknighting for SE by coming to their defense, because I couldn't care less about what you or others feel about the company. The fact is that the statistics gathered in sufficient quantities in the past all point toward there being no quantifiable issue with the systems of RNG in use. Over a statistically significant quantity of attempts, the results trend very closely to the displayed chance.
Fact: it does also spike high. If a system of RNG were only spiking low then the overall result of the RNG would not end up close to the displayed value, yet every test that records statistically significant numbers of attempts shows that it does. Since you end up having the chance equal to the displayed value over a very large number of gathering attempts AND you do get trends of failures, there MUST also be trends of success to accomplish this. So, assuming you believe what you're saying, a few scenarios are possible:
1: Your personal low spikes really haven't been countered so far, and your current results are lower than the displayed value. You're having bad luck. Eventually, you are statistically likely to run into a similar chain of good luck, but you simply haven't yet. It's theoretically possible to never get that high spike to counter the low spike no matter how many attempts you make, but that's possible in the same way that flipping a coin an infinite number of times and always getting heads is possible.
2: Your personal low spikes are being countered, but you just haven't noticed. As I've said before, the human brain is notoriously bad at intuitively grasping probability. People also tend to remember either negative or positive events more than the other, rather than in equal measure. It's more than likely that your brain and memory are working against you in how you perceive your RNG. Alternatively, even if your brain is an enigma that has a perfect intuitive grasp on probability, you may still not have noticed the high spikes if they weren't pronounced enough. You can have extremely strong low spikes, and then otherwise experience a very mild, constant high "spike". For example, you could fail at 50% chance 10 times in a row, then succeed at 50% chance 51% of the time over the next 200 attempts. Both the scenario of bad luck and good luck are "spikes", but the bad luck in this case is far, far more pronounced, and that's the one you'll notice and remember. You're much less likely to even notice that spike of good luck because it's so spread out and doesn't stand out.
These same realities apply to crafting. The RNG is broken in neither. That quote from your FC crafters has been a crafter joke in FFXIV for a long time, but all it's saying is that it's possible to fail unless it's 100% and that you'll just have to accept what you get. Don't take the joke too seriously.
Start reconsidering, then.
Also, it would be in your best interest to stop with the "white knight" nonsense, as your reliance on slander belies your intentions.
Oh! As an aside:
I've gotten HQ without increasing quality at all (1% chance) quite a few times. :P
Hell, I've gotten HQ at all sorts of percentages. 7% is actually one of my more memorable "failures turned wonders". I royally screwed up the rotation (pressed a macro by accident) and didn't have Reclaim slotted. Ended up with 7% Quality, and got an HQ out of it. It was one of those 2-star level 60 recipes for a piece of NIN equipment back in early Heavensward, iirc.
This doesn't prove anything, of course, but it's amusing to think about.