The elephant in the room here is probably that it's kind of hard to really thoroughly shuffle a deck.
Around 12 riffle shuffles will do it for a 60-card deck, with a few more needed for bigger decks. (The complicated expression given in the linked paper is pretty well approximated by 2·log2(n), so doubling the deck size requires two more riffles to achieve thorough randomization.) With three minutes to shuffle the deck between games, that means up to 15 seconds per riffle, which is quite doable with a bit of practice.
But of course, riffle shuffles are kind of hard on cards, so maybe you don't want to subject your deck to a dozen of them before every game. That means resorting to slower and/or less efficient shuffles like the mash (which is actually pretty close to a riffle), the wash (slow and messy, but OK if done well) or even the pile (slow and potentially cheaty) or overhand (just plain lousy) shuffles.
So the upshot of all this is that, unless you shuffle your deck very thoroughly (say, a dozen riffle / mash shuffles) before each game, some traces of the original non-random order are going to remain. The question, then, is whether that non-randomness ends up helping you or harming you — and if you start with a mana-woven deck, it'll probably do more to help than to harm.
In general, as long as everyone starts with a similarly ordered deck and shuffles it equally well, it doesn't really matter if the shuffle is a tiny bit short of perfect, since any advantages or disadvantages due to the imperfect shuffling will be the same for everyone. But if you start with a mana-woven deck and your opponent starts with a non-woven one, it could give you an unfair advantage.