OP, the problem with your argument is that it relies on examples of champions that are banned or picked all the time or not at all. That's not a good argument because picks and bans don't necessarily have anything to do with champion power.
The reason champions like Amumu, Shen, and Blitzcrank are always picked or banned has nothing to do with balance... they're perfectly balanced, if not slightly underpowered when it comes to actual stats. The reason that they are always picked or banned is that they have very simple play that requires specific counter-play. So Blitz throws out grabs. Maybe they hit, maybe not, but unless there's one of a few specific champions on the other team, he's not really punished for trying. Which is fine. And it's pretty easy to avoid letting him have a major impact on the game... ward well, don't travel too predictably when he's around and you don't want to get grabbed, and play behind minions in lane. People ban him because they don't want to have to counter-play him, or they don't trust their team to counter-play him, and they pick him because they hope the enemy team will fail to counter-play him. The exact same thing is true of Shen, Amumu, Malphite, and most of the other champs that people like to cry about because "they're always picked or banned." That is not a reflection of OP.
For similar reasons, a champion not being played often doesn't mean they're underpowered. For instance, Kennen is rarely seen these days, and yet I don't think most people would argue that he's underpowered. He just doesn't fit the types of champion that are played right now. He doesn't do well against bruisers top, and he can't shove lane safely against most of the current mids so he can't roam, which is expected from a mid champion today. That doesn't make him UP. Xerath is in a similar position, although honestly I'm expecting him to be played a bit more with the rise of all AD teams. On the other hand, Cassiopeia isn't played very often... and is definitely not UP. Her problem isn't that she doesn't fit the current gameplay, it's that playing her really well is so complex that very few people have really mastered her, and she's not worth playing without having achieved real mastery.
So, a champion being under-played doesn't mean that they're underpowered. There are a few examples of champions that are honestly just trash, but the only ones I can think of right now are being reworked. Not "will be reworked at some point," but are actually being reworked right now.
And overall, that's where your argument fails. Yeah, I might like to see a few more tweaks going into each patch. And I definitely agree that Riot has recently favored really, blatantly excessive nerfs... nerfs that anybody should have known were too much. But I can't agree with your argument because it assumes that champions that have no real balance problems are unbalanced because of picks/bans, which is inaccurate.