Ahm, first I never even thought once second of Amumu taking a red buff level 1, it's just silly unlesss your own blue is being invaded and you can't risk the invasion.
For Alistar, you should try it, because obviously I did and I ensure you that you can clear the camps you need to hit a decent level without beeing forced back.
You don't need (if mid lane and top helps you.. what they should be doing anyway) to waste your pulverise on the wraith camp.
For the red one, you can use it since anyway you'll regenerate it during the gank etc.
Most of the time you get the kill top or mid anyway so that's not so big of a deal to go back early with an Alistar as you can upgrade boots if you were involved in the firstblood (I also run gold runes/masteries as it should be for this matter). And you DONT, never, ever take gold items first without upgrading your boots with Alistar. So you don't need that much gold to make this back-to-fountain worthy since the +5 boost is just your biggest tool.
I want to underline that I'm not inventing anything there, Alistar played in competitions (even Maokai or things like that) do start red for this purpose, it also optimizes your jungle and with a decent team you seriously don't need blue at all. Else how would you even be able to give it up later since the creeps'hp scale almost as fast as your spells'damages and the mana cost increases aswell so.. Yup items help, but Philosopher Stone wont help you to spam it as it's the only "mana" item you get usually with Alistar.
I don't want to snub you talking about competition and pro players (mostly thinking about snoopeh here) but there is some reasons that make them chose this path.
Besides I main jungle, and the level of the junglers is just horrible. The path aren't optimized, they don't even know the cooldown of the spawning time of the little camps, they don't take timers and things like that, they have no clue what is counter ganking and just blame you for overextending when you dominate instead of keeping tabs on your lane to counter the obvious gank. Jungling isn't about clearing (well depends, but in Alistar case it's sure not the matter), it's about being at the right spot, at the right time. Taking red level 1 ensures a great pressure top lane and this is totally game breaking in normal/ranked games given the early domination provided in a snowball metagame. So yes, don't believe what you see during your games since experienced junglers are rare as diamond (uh this joke).
For Renekton, ahm it has really not much in common with Udyr.. like Udyr ganks are almost always relying on your opponent being outpositionned.
Do you realise Renek got almost the biggest gap closer (and it's not a skill shot like Lee Sin's q) and yes there is this stun like Udyr. But that's all, the difference between Udyr speed boost and Renekton Slice and Dice is really massiv.
Let's just say that Udyr cleans quite faster, but in early game you rather have a Renekton in your jungle for the ganks that virtually Udyr just can't take given Flash. With Renekton you bridge the gap in a second, with Udyr you need to run after the enemy that's really not easy to get him this way if he's not totally overextending. I'd say the thing that comes near to Renekton would be Pantheon, or Xin maybe..
And no, by the way it's not "low level" or gimmick to not pick the usual junglers. You know, Mundo waited a LONG time before being picked (even after the buff actually) and there were some top ranked players that were playing it before everybody think it's viable. If I remember correclty it was also a player that mained Twitch jungle (before the shift) and at this time when you picked it (Mundo or Twitch) people were just like "Oh my god stop trolling imma report".
Same goes for Ezreal in fact, a top corean player (I'm not sure) JUST plays him all the time, either mid (the greatest Karthus counter is there by the way), jungle or bot lane obviously.
So yes, it's not only for fun, it really works but obviously if people keep playing the same thing over and over noone will realise it. Just saying, for instance the fastest jungler is Olaf, and almost noone play it, and this is nonsense. People just rush Lee Sin (and for good reason, it's an insane pick) but can not adapt their pick to the role they'll need to fulfill.
Oh and yes, you're totally right on the counter jungling part of Karthus, but if you can "outplay" your enemy jungler this can be overcame. But yes, this is true, you better make sure your team is decent enough to provide you the cover you need so that the enemy jungler wont counter you all day (by the way Ezreal is an insane counter jungler once he gets Wriggle with his E that provides an easy escape and his huge damage on a single target with the ability to proc wriggle faster than anyone).