I've yet to see a champion fighter sword and board mentioned as beating the monks damage.
Champion is one of the worst fighter subclasses, and sword and board isn't particularly well suited to the features they get (or damage, generally, due to lack of feat support). And indeed by a quick back-of-the-envelope estimate, they fall behind the monk for at least a few levels in late tier two, when the monk can be flurrying all the time and the fighter hasn't gotten their third attack yet. But from level 11 on it looks like they're back ahead of the flurry-always monk, though not necessarily by a lot. That said, I haven't factored in indirect damage such as stuns, or shield master prone, etc. Right now the champion is sitting there with their bonus action completely unused except for the occasional second wind.
Though if you just replace the longsword with a spear or quarterstaff and take Polearm Master, it's a different story, though Champion still isn't really adding anything to that over base fighter.
For Paladins are you assuming they get to use all their divine smites? A daily resource.
Speaking of daily resources, how are we measuring a monks at-will damage against a bard? Do the bards get their daily spells? Or are they sticking with at-will damage too?
As you can see, I'm skeptical about ANY class.
At-will damage is the wrong measure, even for damage. At a minimum you should look at either something like 'damage output in a typical adventuring day', though even that undervalues PCs with burst abilities.
I don't think anybody is saying the problem with monks is what they do when they're out of ki, necessarily. The problem is
that (a) in order to make their turns using ki reasonably effective it causes them to spend too much of their time out of ki, and (b) even when they're burning through ki at a fast pace, they're doing less than other characters do when they nova.