Oh hell yes they belong on the sidewalks!
First off, I hate running across bikers on the road. I have to dodge them in such a way as to make sure I've given them enough room, not just so that I don't hit them, but so that I don't risk making them lose control, and in doing so I risk squeezing too close to any cars in the lanes on the other side of my car... that or slow down until I have an opportunity to pass them, often holding up an entire fucking line of cars all stuck behind a bike which moves much slower than we want to be moving. And now that I've heard the harrowing tale of what it's like to be the one on the bike, I'm even more opposed. No one should have to feel like they could make one tiny mistake and DIE for the entirety of their trip to work/school EVERY DAY! That's the kind of stress that'll give you like twenty fucking ulcers!
Secondly, bikes on the sidewalks are not a problem at all. I happen to have actually been hit by someone on a bicycle once, and you know what, it really wasn't all that bad. I mean sure, if I were frail it could be a problem, but it really doesn't cause any risk to the vast majority of people. And hell, I only got hit one time, out of the entirety of my twenty one years here on earth, hit once. And I haven't actually talked to a lot of people who have been accidentally hit by someone on a bike while walking, so the statistics are probably really low that you get hit.
Bicycles are infinitely less upsetting to the normal operations of things when they're on sidewalks, and that's where they need to stay!