I noticed that your astute readers picked up on the fact that we lovingly pilfered the "vs. mode" idea from Croal/Totilo's exchanges in no time flat. It bears mentioning that we stand on the shoulders of giants, etc.
It's funny to me that you described parts of your experience as drudgery. I was on a console-less vacation last week and one of the first tasks for me when I got back to my 360 yesterday was finally cracking the last few segments of Braid-- my girlfriend was my sidekick and puzzle-solving assistant as I played through the latter half of the game, and we were joking that it was my "homework." I think part of this feeling comes from the way that blogging colors my experience (maybe you felt this too, where there is this pressure to finish the game in order to become well-informed), but it also speaks to the ethos of the game itself-- puzzling my way though Braid was onerous. Blow himself is quasi-puritanical about making the player work hard to figure out things on her own, and this attitude is reflected in the game's design.
The positive side of this, as I see it, is the sense of earned satisfaction you get from mastering the new rules and teasing out the diverse logics of the game-worlds. And it as it happens, this is exactly the sort of thing I appreciate. With a the exception of few puzzles ("crossing the gap" on world 5, for example), which I couldn't have gotten through without just stumbling onto the right solution, I felt that the demands Braid makes on the player are reasonable-- uncompromising and rigorous, but reasonable. Braid is an imperious mistress, but she is rarely fickle.
The negative side of this, as your experience illustrates, is that Braid just lacks any immediate sense of fun. It does not set out to entertain you, and with the exception of some pretty aesthetic moments it makes you earn the pleasure you take from it. (Portal, which makes for a good point of comparison, wants the player to like it and desires to be understood in a way that Braid does not.) I think part of this is that the feel of the platforming is kind of stiff compared to contemporary platformers-- the fact that it was sometimes difficult to execute the proper solution to a puzzle because you couldn't jump properly is a design flaw, in my opinion, and imposes needless barriers to the core enjoyment of experimenting and problem-solving. (Part of the problem here is just that Nintendo makes everyone else look bad when it comes to making buttery-smooth and tactile platforming controls.) This is too bad, because I liked the fact that the rewind mechanic removed the need for the frustrating-controller-slamming-repetitive-death and platform-pixel-length-estimation that is endemic to platformers (indeed, I think this idea of removing player death from the rule-learning scenario was one of the best ideas in the whole game design); it was unfortunate that some of the unrewindable elements in the later puzzles made the easy "redo" impossible.
You're right that the deep concentration and tricky jumping you have to perform to solve the puzzles pulls you out of the narrative. I'm not sure if I would say it "clashes" with the narrative-- as you say, there are some interesting and complex thematic connections between the the texture of Braid's play and the narrative elements. (I have some theses-- crackpot theses-- on this front, which I will inflict on the internet at some future date.)
So I feel where you're coming from Mike, and I understand your disappointment. I thought there was something really refreshing, even respectful about the way Braid makes a set of stringent demands on the player. It shows a certain confidence in the player's capacities. But while meeting these demands can feel ennobling, it can also be alienating to play a game so rigorously governed by the intentions of its author. Braid offers a very specific type of fun, and if this sort of puzzling doesn't suit your temperament it's a game that's easier to admire than to love.