> Does the faster player get nothing in return for their speed advantage?
In Guidelines Tetris, you only deal damage if you do a sufficiently complicated maneuver. A Tetris (4-line clear), a T-Spin ("spinning" a T-block into place), or a "Combo" (repeatedly clearing lines non-stop). As such, you only can do a major damage from I-blocks (a 4-line clear / Tetris), or T-blocks (TSpins).
No other block in the game can deal major damage by themselves. So the placement of I blocks and T-blocks is fundamental.
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Also note: Garbage is a "comeback" mechanism: large amounts of garbage "feed" the opponent lines to perform a combo with. If you deal non-lethal damage to the opponent, it is common for the opponent to use the garbage you sent him to "combo" into a large amount of damage back to you.
As such: the game becomes more garbage is sent around. There is a chance the opponent can see a 12-combo through the garbage you sent them. Its difficult for the untrained players to see combos, but it definitely comes into play at higher-levels.
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The current system (including the bag-randomizer) has been carefully balanced between many issues, and playtested for years. Aside from occasional issues (ex: TSpin singles becoming too easy in an early version: leading to the development of "TSpin Minis" to nerf that technique), the overall Tetris Guidelines rules are quite good. I recommend playing a modern competitive version of Tetris and trying it out yourself.
Combo-play is definitely considered one of the strongest strategies right now. Moving to a true-randomizer will necessarily nerf Tetris / T-Spin play, because you can no longer predictably get Tetris or T-blocks.
In my experience, most players want to play Tetris: that is, create Tetrises (4-line clears). Combo play is certainly fun, but it makes sense to "buff" the canonical "Tetris" strategy of the game since that's what people expect. Tetris is the name of the game after-all, and you can only play "Tetris" if you have some kind of assurances that I-blocks are on the way.
Nerfing Tetris and buffing Combos seems to be the wrong direction for balance. If anything, Combos need to be nerfed.
Perfect-clear play is also lovely to watch. Computers are better than humans, but when a human perfect-clear player comes in, its truly intense and fun to watch (and play against). Perfect-clear play relies upon the bag-randomizer to a severe degree, and isn't even that strong of a strategy (a good perfect-clear player needs to threaten 3x perfect clears within 20 seconds to be competitive at high levels). Moving to "true random" would erase this obscure style of play completely.
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In any case, there's a big community which prefers to play with true randomizers: the "Tetris Classic players". They play by a separate set of rules entirely.
As a Guidelines Tetris player, I obviously have my bias. I think the game is more fun with Tetris, Combos, and Perfect-clear openings in play.