The short answer is "No." No-one knows how to do that.

The longer answer is "It's complicated." Processing power, crystalized knowledge, and rationality are three different things.

Processing power is probably mostly genetic in humans and there's not much you can do about it. Exercise, nutrition, and nootropics can help, but only to a degree. Learning different tactics to apply to problem solving or mnemonics can help you use what you have more effectively. Still, the human brain is remarkably plastic. If I had to guess, techniques like sensory substitution, biofeedback, or advanced meditation might let someone reallocate their gray matter and improve their processing power that way. But no-one knows how, so that would be original research.

It takes a surprising amount of processing power to beat expert knowledge. When Kasparov lost the chess match to Deep Blue in 1997, he still won two games in the match! A lot of "smarts" is knowledge. A large language model takes a lot more processing power to train than to run inference. There are limited situations where figuring everything out on the spot is necessary. More processing power does let you acquire that crystalized knowledge faster, so those with higher IQs tend to have more of it, but so does effort, allocated time, effective teachers (private tutors), and accumulated civilizational knowledge. You can catch up with and even surpass a higher-IQ individual who isn't really trying in terms of crystalized knowledge if you take full advantage of these other factors.

The capacity for effort is partly innate talent, but nootropics can probably help you here more than for raw IQ. You probably have as much time as anyone else, but if you're willing to sacrifice time spent on other things, you can dedicate more of it to study. If you can afford private tutors, they're much more effective than classes. If you can't, you can ask GPT-4 to pretend to be one, and that might still be more effective than classes. There has never been a time with more available knowledge. You can read textbooks.

And finally, the sanity waterline is pretty low. Rationality can help you avoid a lot of stupid mistakes. Who you hang out with really affects how you think. You don't need a super-high IQ to learn and practice rationality. Most of their edge would go into finding us in the first place. But you're already here.

Maybe a slightly pity answer, but comparative advantage is worth learning about and Being the (Pareto) Best in the World is worth a read.

It would help if you clarified why specifically you feel unintelligent. Given your writing style: ability to distill concerns, compare abstract concepts and communicate clearly, I'd wager you are intelligent. Could it be imposter syndrome?

You can compete with someone more intelligent but less hard working by being more organized, disciplined, focused and hard working. And being open to improvement.

diet, exercise, a disciplined schedule

Do you also get enough sleep? (What is your definition of "disciplined schedule"? For some people it might mean: going to bed early to get enough sleep. For other people it might mean: do not waste time doing unproductive things such as getting enough sleep. The latter would be harmful.)

the mental fog remains

What exactly do you mean by this? Is it difficult for you to focus on things? (Maybe you have ADHD, or maybe you are just too distracted thinking endlessly about how "stupid" you are.) Or is it difficult to understand some complex topics, because you haven't learned the simple ones yet? (Start with the fundamentals, then. Read a textbook.)

And what about education? That’s supposed to be the great equalizer, right?

In theory, things like Khan Academy are equally accessible to everyone. (If you have a computer. If it is translated to your language. Assuming optimistically that you have enough free time without interruptions.) But the existing school system is very far from this ideal, for various reasons:

• many schools suck, many teachers suck;
• the education is designed for the (below-)average student, so the smart ones either succeed to get to one of the better schools or are ignored by the system;
• there are many artificial obstacles in the system, for example you have to learn things at the specific time, and if you can't (e.g. if you are too sick, or can't focus on learning because your parents are divorcing or you are bullied at school) you won't get a second chance;
• generally, money can buy success, by getting you to a better school, by hiring tutors, by not needing a job after school hours, etc.

So it is also possible to conclude cynically that the purpose of education is to legitimize the existing inequality in the society -- to convince the people at the bottom of the social ladder that they deserve it because they are inferior ("stupid", "lazy", etc.).

The current state of education seems more geared toward churning out graduates rather than fostering genuine intellectual growth, especially for those who don't naturally excel.

Yep, as long as the most talented ones succeed to learn, there is not much motivation to make it easier for the less talented ones. (Sometimes there is even active opposition. Education system has two major tasks: to teach some people, and to separate the smart from the stupid. If it does too good job at the first task and accidentally teaches everyone, it means it has failed at the second task.)

As for intelligence being multifaceted, I understand the arguments. Yet, at its core, there seems to be a singular, critical capacity for understanding, learning, and problem-solving

Exactly, that's what generally referred to as "IQ".

Educational Software: These keep me engaged, but do they make me smarter?

Software cannot increase your IQ. But it can teach you things. Understanding things is one of the reasons why you want to be "smart", right?

It suggests that the relationship between intelligence, as we measure it, and success is more complicated than we'd like to admit.

Success = innate abilities + work + privilege + luck.

IQ contributes a lot to the result, but is only one of multiple components.

I’m seeking substantial, proven methods to increase my cognitive capacity.

To increase IQ, there is no known way. If you are healthy, get enough sleep, and checked for potential problems (including allergies or anemia), you probably did what you could.

Accumulating knowledge takes time. You mentioned nothing about what you study and how you study. Perhaps you are making some mistake there. Maybe you are watching YouTube videos or reading popular science magazines, instead of reading textbooks and doing online courses.

For problem-solving in particular, I've really enjoyed how this youtuber articulates specific strategies.

Why do you want greater intellect -- what do you want to do with it?

Consider how to get directly to those ends with the tools you have available. Consider the option of disregarding the abstract concept of "intellect" entirely, and simply going out and doing whatever it is you're waiting to do until you "get it", modifying your approach after each mistake.

Often when those around me describe something I said or did as intelligent, it actually feels from inside as if I simply identified my goal and went straight toward it rather than getting hung up on the expectations about what was or wasn't "supposed to" work. The insights and solutions that seem intelligent to onlookers come less from having some ephemeral "iq" trait and more from my willingness to notice the irrelevance of imagined constraints and disregard them.

Also, if you reflect on your attempts to learn things, it refines the meta-skill of learning stuff. I've been surprised by my ability to learn things I'd previously assumed I'd hit the intellectual wall on when I revisit them after having had more experiences of learning things in different ways. Avoid the trap of assuming that failing to grasp something quickly in one situation means you'll never grasp it quickly in another.

My approach to the personal stupidity issue is the "additive nature of intelligence" theory. It suggests that by spending more time on a problem or by seeking assistance from others (like AI, search engines, or people) and using tools (writing, drawing, etc.), you can achieve the same outcomes as a smarter individual.

This concept also posits that certain ways of thinking, such as Descartes' method, can yield notably good results. 

(My day-to-day job is literally to tackle the 'generality' of intelligence)

While having high IQ/g is useful, it is not what lies at the core of great performance. Having developed 'intelligences' around the task you're tackling, + determination/commitment/obsession, + agency is what creates great results. 

I think it's better to focus on things one could change/train, sadly IQ/g is not one those things.