From a design point of view, I cannot figure out why Cascade should specifically exile the cards involved, then return them to the bottom of the deck instead of revealing them (like Spellshift or Reweave do)? Is there some sort of interaction I didn't notice?
Cascade works with exile rather than revealing because of the possibility of one Cascade triggering another Cascade.
The full meaning of Cascade is (702.84a):
When you cast this spell, exile cards from the top of your library until you exile a nonland card whose converted mana cost is less than this spell's converted mana cost. You may cast that card without paying its mana cost. Then put all cards exiled this way that weren't cast on the bottom of your library in a random order.
With this definition, if one Cascade spell finds another Cascade spell, the way you handle it is easy: the cards exiled by the first Cascade are still exiled, so you keep going through your library until you find another appropriate spell. It's simple and a very literal cascade through your library.
What if you were simply revealing cards from the top of your library? Well, the "reveal" doesn't actually move the cards from your library. So if you needed to Cascade again, you have to put those cards back and then look through them again! Then, making the "put all cards revealed this way" phrasing work becomes much more complicated, because some of the cards are subject to two or more Cascade effects.
In short, I don't believe that there's a deep gameplay reason, but it makes the Cascade mechanism (specifically, its recursive nature) and the wording simple and elegant.
As far as I can tell, there are two functional differences from cascade exiling cards instead of merely revealing them:
It lets you cast cascaded spells from exile rather than your library, circumventing Grafdigger's Cage.
The exiled cards are not considered if the abilities of cards like Selvala, Explorer Returned, Charmed Pendant, Millikin, or Deranged Assistant are used to pay for an additional cost on the cascaded spell. These abilities are rare in that they are mana abilities (and so can be used during the resolution of another ability) that interact with the top card of your library.
There seem to be several types of effects that are looking for a top card of your library that meets some criteria.
I can find one example, namely Jalira, Master Polymorphist, that has you reveal cards off the top and put the revealed cards on the bottom in a random order. There are also Heroes' Podium, Aladdin's Lamp, that put unrevealed cards on the bottom of your library in a random order.
Finally, Ripple seems to be a similar ability to cascade, but uses reveal instead of exile and doesn't shuffle the cards before they go on the bottom of your library:
Whenever you cast a spell, you may reveal the top four cards of your library. You may cast any revealed cards with the same name as the spell without paying their mana costs. Put the rest on the bottom of your library.
So, as far as I can tell from this, the cards are exiled because you are casting one of them, shuffling the rest, and then putting them on the bottom of your library. Given that there are potentially a lot of things that happen when you cast a card (such as choosing targets, paying additional costs, etc), having the cards in a dedicated area before they are randomized seems like a reasonable idea.
The most comparable card is Possibility Storm, which uses the same procedure as cascade. None of the other cards that have you shuffle cards from the top of your library before putting them on the bottom of your library have you cast anything (see: http://magiccards.info/query?q=o%3A%22random+order%22+-o%3Acascade&v=card&s=cname).
Why this procedure isn't used for Spellshift and ripple is anyone's guess. My speculation is that Wizards was hesitant to have abilities that shuffled a set of library cards independent of the library, given that only Aladdin's Lamp (excusable as an ancient card) did this until 2014 (when they printed Heroes' Podium).
NOTE: The answer by sitnaltax (https://boardgames.stackexchange.com/a/12671/9999) is factually incorrect. If you cascade into another spell with cascade, the second cascade trigger does not resolve until the first cascade trigger is finished resolving (namely the cards exiled are put on the bottom of your library). This is because cascade is a triggered ability. The relevant rules are the following:
603.2. Whenever a game event or game state matches a triggered ability's trigger event, that ability automatically triggers. The ability doesn't do anything at this point.
603.3. Once an ability has triggered, its controller puts it on the stack as an object that's not a card the next time a player would receive priority.
No player gets priority until the first cascade trigger is done resolving.