I have never run a single-class campaign (defined as one in which all the PCs are of the same class: everything is magic-users; everything is paladins; everything is thieves.) But I would like to.
I suppose you could call me single-class-campaign-curious.
The interesting thing about the single-class campaign is that, in closing off many old possibilities, it opens up new ones. A campaign in which all the PCs are magic-users is not going to involve a lot of combat. It is going to be one in which they have to use their wits to get by. It is going to revolve around searching for spellbooks and scrolls as much as treasure. A campaign in which all the PCs are paladins is going to involve heroism and unreconstructed good-guy antics (because the PCs are going to have to stay paladins). A campaign in which all the PCs are thieves is going to involve a lot of sneakery, deception and plotting, because that is how the PCs are going to get treasure and hence XP. One in which they are all fighters is going to be combat-intensive, but will also revolve around resource management and good relations with NPC healers. And so on.
The tone of the single-class campaign partly derives from mechanical incentives, of course. An all magic-user campaign will naturally be low on combat because the players will avoid it - their PCs are fragile. But it will also derive from tone. D&D is a roleplaying game; if the PCs are all paladins (or whatever), the players will act accordingly.
As well as setting their own specific tones, single-class campaigns would also seem to provide their own justifications and origins. The magic-users are all members of a secret cabal; the thieves are a Heat/Ocean's Eleven style joint criminal enterprise; the paladins are a band of holy avengers cleansing the world of evil; the fighters are a band of warriors trying to get by in a high-magic world; the clerics are all members of the same faith, seeking to gather disciples; and so on.
The only class that I would find too cringeworthy for a single-class campaign is bards. A "band" of wandering minstrels is just too on the nose for me and too ripe for jokes and Mick Jagger impressions - as long term readers will recall, I'm firmly of the view that D&D has to be the straight man.