In his Fighter vs Wizard video, he does a good job of outlining the history of the issue, and goes into a little bit of the actual rules from Pathfinder and/or 3.5 edition. A big point that he goes over is how the fighter plays very different than the way a wizard plays. He is emphasizing that the two classes play differently, and those two classes appeal to two different types of gamers. If the game provides the avenue of appealing to (and hopefully satisfying) different mindsets at the gaming table, then balance should not be the big issue in the debate.
It is a fair point, and one that I would characterize as if you are having fun at your table then who cares what people say. But on the opposite side, if there is something not fun about the game at your table, then it is great to bend or add some rules and see how it goes. If you look at the history of the game, there is a trend to give fighters more. In Unearthed Arcana for AD&D, fighters started getting weapon specialization. In the Masters Set for BD&D, there was similar weapons mastery system that the fighter excelled in.
I think another complaint against the 4th edition use of spell like powers for fighters, is how the hegemony of the power system inherent in every class makes the classes seem bland in comparison to each other. 4th edition mimics MMOs in quite a few ways, but the one area they seemed to have missed the mark on was differentiating the resource management systems of the different classes. In the quest to provide combat parity between the fighter and the magic user, they utilized the same powers per encounter system.
There was another book, similar to the Book of Nine Swords, that received excellent reviews that also addressed the fighter versus magic user balancing act (although, in all probability, that was not their primary goal), Iron Heroes takes the approach of limiting what a spellcaster can do, and also boosts what a fighter can accomplish. But more importantly, it accomplishes this task by having different resource systems in place for the two classes.
Now that the next edition of Dungeons and Dragons is on the horizon, it appears as though they are going to go back to having a Vancian fire and forget spell system. Perhaps the designers realize there is appeal to having different mechanics for different classes. Hopefully, the pendulum does not swing back to far in the other direction and we begin right back where we started.