I definitely agree with both Silman and Avni. In fact, so does de la Maza! When you remove all the self-advertising from his book, you will see that he himself studied complete games, studied endgames, studied positional chess. His weakness was tactics. Targetted training is a well known, and often recommended technique.
Michael de la Maza's method isn't earth shattering. His book is like so many self-help books in that it contains a kernel of well-established information and smothers it in self-congratulatory advertising.
I once read Korchnoi commenting on missing a basic tactic. He said he would have to spend two hours a day for the next week working on short tactics puzzles. Botvinnik said that he was weak in "petit combinations", and so worked hard to improve on those. Kasparov has said that each player should start by evaluating their weakest area and work to improve that.
Working on tactics is necessary for chess improvement, but is not sufficient by itself to guarantee improvement. Some players will benefit more from de la Maza's book than others, but only if they are relatively weak on tactics to begin with.
"Every now and then someone advances the idea that one may gain success in chess by using shortcuts. 'Chess is 99% tactics' - proclaims one expert, suggesting that strategic understanding is overrated; 'Improvement in chess is all about opening knowledge' - declares another. A third self-appointed authority asserts that a thorough knowledge of endings is the key to becoming a master; while his expert-friend is puzzled by the mere thought that a player can achieve anything at all without championing pawn structures.
To me, such statements seem futile. You can't hope to gain mastery of any subject by specializing in only parts of it. A complete player must master a complete game ..." - FM Amatzia Avni (2007)