There are a couple things at work here. The main thing to realize is that Ranked Solo/Duo queue gives you a division and LP, and that is meant to measure your progress as a League of Legends player. You used to have a visible Elo. Whenever you won, it went up. Whenever you lost, it went down. If you hit 2000 Elo, it had no permanence. Losing a game would put you back under 2000. There was nothing special about crossing that threshold. The introduction of the LP system was meant to add some tension and relief, and make a handful of games feel more meaningful. You'd stress more about hitting a threshold and then you'd get to experience relief from hitting a milestone that wouldn't just go away on your next loss.
It's a little like the Mastery system for champions. If you play a champion for the first time and you go 19/0/24 with 500 cs, you don't get Mastery level 5. If you are on your main and you go 0/7/7 with barely any cs, you don't drop back down to Mastery level 1. It is a measure of your progress, not how well you just performed today.
In order for divisions to accomplish the same thing, they can't be directly related your MMR, which attempts to reflect your exact skill level (or odds of winning, if you prefer to think of it that way). If it did, they would have to take away your progress as soon as you started losing, and that runs counter to the whole point of introducing divisions and tiers. So that's why your division and your MMR can drift apart. MMR is for measuring your skill level to make fun and fair matches, division is for giving you a sense of progression and accomplishment.
They could use your division for matchmaking instead of using your MMR—like you suggested in your 2nd paragraph—but this would result in less fun and fair matches. There would be more games where some people are just significantly better than others. MMR is a better method of putting people on teams because it is a more accurate estimation of your skill.
And the Silver Vs that you're seeing do not necessarily have Silver V MMR. If you are in Bronze II and you have Bronze II MMR, you are fighting against Silver Vs who also have Bronze II MMR.
