The thing is, there are more than enough great climbs in each country to populate more than one quality GT route. Therefore you don't need to keep reusing the same ones. Or if you do, shake up what you do with them.
Climbs like Alpe d'Huez and Lagos de Covadonga are now ingrained into the Tour/Vuelta fan's mind. They do them regularly. But we still dream of Mont Ventoux, which is a much rarer proposition. The Stelvio, as mentioned, has only been used 6 times in the Giro's history.
I understand the argument that a climb needs to be used regularly close to its inception in order to establish its reputation - indeed Angliru and Zoncolán are testament to this. But once the reputation is there, it can only be diminished by regular use. If there's a chance to win there every year or two, the value of the win there diminishes because it's open to more people.
Fedaia, as mentioned above, is a regularly used but highly regarded Giro pass. It has only been used as a finish once (2008); that made it seem quite special. It's one of my absolute favourite climbs. Tre Cime di Lavaredo is seldom used. Sure, it doesn't have the reputation of a Mortirolo, but it makes it seem valuable to win there.
Taking a climb out of the main event doesn't prevent its use of course, it can still be used elsewise - the Dauphiné using Alpe d'Huez this year, for example. Similarly, if the Vuelta went a few years without Lagos de Covadonga, then maybe the Vuelta a Asturias could use it instead of using the same climbs year on year (the Alto del Acebo is invariably the queen stage), and that could in turn give that race some extra credibility in terms of the field taking part.
Also, there are celebrated climbs going unused in the GTs, for a variety of reasons. Lagunas de Neila, the key climb of the Vuelta a Burgos, is a great example - always entertaining, an excellent climb, but no Vuelta. New climbs are being found regularly, so why do we have to rely on the same old ones?
The answer, of course, is that the GT organisers are trying to sell their GT. They want to sell the riders on it (and riders will want to get wins on these mythical mountains, and be the first on new mountains), and they want to sell the press and fans on it. For many fans, they may only know of the legendary climbs, and so it's easier to sell them on a Ventoux or Alpe d'Huez finish. But instead of Huez, why not use the less commonly used Les-Deux-Alpes once in a while?
You can shake things up without creating a farce like the '09 Tour route. The 2004 Giro attempted it; and that was probably a failure too. The stage over the Tonale and Gavia to Bormio 2000 would have been great if it had just been about 40-50km longer - get the legs more tired before they get there.
I like the decision of the Giro to jump across into Austria; but why the Großglockner and not the Rettenbachferner? That's only been used twice in total, both in the Deutschlandtour, and the lack of use could make it more valuable; the 'glockner is crossed every year in the Österreichrundfahrt. Perhaps the Vuelta could consider hopping across the border into Portugal once in a while, I'd love to see the Vuelta climb to Torre.
One of the problems with the Tour is that it tends to be formulaic in route, because the country's big mountains are in two specific areas and the way the race ends is seen as sacrosanct. But why should it be? The Vuelta always ends in Madrid, sure, but it has mountains overlooking it; it could finish with a mountain or a TT preceding the final procession. The Giro moves around suiting its needs. Why could the Tour not do the Champs-Elysées earlier on, or start there, and finish in Lyon, allowing for the mountains to run right to the end like in the Giro?
The thing is, if the racing is good the mountain doesn't matter. The racing on Pal (PAL!) this year in the Vuelta was as good as the racing in more storied Tour de France climbs. If the Tour is coming to a close finish and people need to attack then it won't matter if they're climbing to Courchevel or Arette-Saint-Martin instead of Huez or Tourmalet.
The main thing is, the Grand Tours need these mythical climbs, for the legends to be born and made. But they don't need all of them all the time, if they create routes that allow for either a) new mythical climbs to be created, eg Bola del Mundo or Zoncolán, or b) the racing to be remembered far beyond the race itself - and they don't need the mythical climbs for that. However, what these mythical climbs do is offer the race organisers an easy out. They give the race a feeling of grandeur without the organisers having to think too hard.