If Silman really said that you should start with just a month of tactics, he is wrong or some context is missing. Some people have more of a natural gift for tactics, so perhaps they can do fewer tactics and get the same benefit as someone who does many, but if anyone below the level of, say, A or expert wants to increase his ratings rapidly, tactical problem solving is where it's at, and you really can't do too much of it. You will find that with all of your book reading that your OTB rating will inch forward and probably come to a complete halt at some point under your desired goal if you have even semi-lofty ambitions.
As for the assertion that if two people have the same tactical ability, having better positional knowledge will win. Well, of course! However, if two people of equal natural ability study chess and one does 70% tactics and 30% other stuff, while the other does 30% tactics and 70% other stuff, the guy doing the tactics will be a much stronger player after one year of study than the guy who was not focused on tactical training. I haven't done a scientific study, but I see evidence of this in my son's progress as well as the rapid progress of the latest generation of young players, who do much more tactics as well as much more blitz and rapid play thanks to the advent of computers and the Internet.
Think about it another way. I think it was Dan Heisman who said that a computer that looks three moves ahead but has zero positional knowledge will almost always beat a computer that looks just two moves ahead but has perfect positional knowledge.
I agree that tactics is not just about calculation. It is about:
1. Calculation
2. Recall
3. Intuition
Recall is the ability to spot tactics without calculation because the tactical patterns have been ingrained in long-term memory. Tactical training with puzzles as well as lots of gameplay will get the job done here. The problem with correspondence chess is that unless you're playing hundreds of games simultaneously you're not going to get to see a whole lot of tactical patterns over a short period of time. That's why I recommend blitz as well as puzzle solving instead. Also, to help this stuff get into long-term memory, do repetitive solving...solving the same positions multiple times. Michael de la Maza recommends taking 1000+ puzzles and solving them seven times each.
Intuition is the ability to 'see' the best move without calculation and without conscious recall of the same position from another game. Some people confuse chess intuition with positional or strategic knowledge. Some of that does play a role, but the main component of chess intuition is the large databank of positions you've seen and solved for in the past. Your brain has processed the position or something like it before so it knows the final answer even though it may not recall how it originally arrived at the final answer. Magnus Carlsen recommends lots of blitz to help build up your chess intution.
Not to say correspondence play is bad. It'll definitely help you have patience in your games. However, if you don't balance this with regular time controls so that you can handle thinking while on the clock as well as rapid/blitz play, so you can ingrain lots of real-life positions and problems in your brain, you're not going to become a good player at standard time controls.
If you're not rated 1800/2000 yet (USCF OTB, not correspondence), you need to focus 80% of your chess study on tactics and calculation. If you're reading books regularly, you're just not going to be spending enough time on tactics.
I did the traditional book route. Like Senchean, I got pretty rapidly to 1500 but then I got stuck at USCF 1500 for years (decades really). My son is doing tactics and tactics only, and he's zipped by me in less than a year after learning how to move the pieces. He will be an A player very soon (maybe this week, maybe next month).
Senchean, you said it yourself: "The brain cannot learn something it does not already know." Chess at its foundation is tactics. The ability to both spot and calculate tactics is the fundamental skill you need before you can advance, before you should tackle openings or strategy. Endgames are useful in that they can help develop calculation/tactical ability as many endgames are just tactical exercises with a smaller number of pieces. And it's not just a few months of tactics you need upfront, it's years and years of the stuff, particularly if you're an adult and won't absorb that material as quickly as a young'un.
Well, a couple of things. The suggestion of spending a month on tactics, was made by IM Jeremy Silman, and it's just a place to start for beginners.
Second. I spend at least a half an hour on tactics every day using Tactics Trainer on Chess.com and it has helped a lot. But the idea that tactics is the foundation of chess I feel is fundamentally wrong. You can practice tactics all day long. But if your pieces are never in a positon to pull off those tactics, then the tactics training doesn't mean a damn thing. This is because you can win a game based purely on the buildup of small advantages without every using a tactic besides mate. But you cannot win a game without putting your pieces in the right position. And I do feel, all other things being equal, that a person rooted in positonal understanding facing someone who is a tactical player of equal strength, the positional player will win 9 times out of ten. I know the saying, "Chess is 99% tactics." But It's taken out of context. This is because tactics training doesn't just help you find tactics and pull them off over the board. They help you see your opponent's tactics and prevent them. Then the next level is to use tactical shots to achieve positional advantages.
Finally, Botvinnik said, "Tactics are the Servants of Strategy." And that is the real purpose of tactics, to help you achieve strategic goals. This is as true over the chessboard as it is in actual warfare. And in chess, you have have to have a plan from the first move. But you usually cannot pull off tactics, unless someone makes a huge blunder, until pieces start having influence over the opponent's side of the board. But it is also true that I have a highly strategic bent. It sounds like your son is much more of an attacking player than I am. All I know is, this program is working for me.
And as far as calculation, you can use books to learn calculation and visualization. First, most people have an incorrect idea about how calculation and chess work. They think they have to calculate 10-20 moves ahead in order to play well. This is just false. So long as it is accurate, you only have to calculate 2 and a half moves ahead. That's all you need. GM Khachiyan calls them Bushes. I learned about this false assumption and the correct way to think about it by reading books, specifically The Inner Game by Andy Soltis.
Second. When I read a book, I put the actual moves of a game into my computer, but when a variation is mentioned, I calcualte it in my head. This has helped me be able to calculate up to 5 moves ahead accurately so far. When I feel comfortable with it I'll move on to 6.