It's been quite a while since I've read anything strongly touting unified mechanics in RPGs. That could be because most of my reading is in OSR circles, where most people seem content with, or at least tolerant of, the idea of a game of heterogenous mechanics. Even so, I still see occasional mentions of it in passing, and there have been musings on the incongruity of the percentile thief skills, attempts to render character abilities on d20, etc. Sometimes there's a good reason for rejiggering something to use a different die or combination of dice, but in my opinion, simply unifying mechanics is not one of them.
For my part, I like and prefer the diversified mechanics of old school D&D. Firstly, not all mechanics work the same. Sometimes you want the anything-can-happen linear randomness of a d20 or d100. Other times, the neat bell curve of a 2d6 roll, with results weighted toward the middle, models things more aptly. For very simple and routine tasks like opening stuck doors, surprise, and initiative, a simple 1d6 roll makes sense. Some mechanics, such as the spell-caster's spell slots or a simple comparison of an ability score to a target number (like a boulder that requires 20 points of strength to move), use no dice rolls at all.
Secondly, it makes different tasks feel, well, different. A unified mechanic, which by definition resolves every action with the same type of die roll, doesn't accomplish that. I know, it's all about imagining what the roll represents, but in a game in which the roll itself is the most immediate connection between the player and the character's action, it necessarily influences the players' perceptions of what's happening. Picking up a different die or dice feels like taking a different tool out of the toolbox.
Some might argue that a unified mechanic simplifies play, but some of the editions that attempt unification of mechanics are wildly complex, while B/X is noted for its relative simplicity and ease of play.
In a video game, where all the number-crunching happens out of the player's sight, the particular mechanics don't matter at all so long as they produce a result that satisfies the player's expectations. In a tabletop RPG, though, not only does it not bother me that physical combat, the thief's skills, the cleric's turning ability, and the magic-user's spell casting don't use the same dice as each other, I prefer it that way.