What exactly is the point of the "capstone" features? Unless your group goes beyond level 20, won't your campaign be over, or virtually over? Most of the capstones are incredibly awesome, so it would stink to not get to use them.
First, your level doesn't determine when the campaign ends, the story does. You could reach level 20 while in the final leg of the campaign, but before the end. So you could use those awesome abilities in the final battle. Heck, you could reach level 20 before the final arc of the campaign, and have a while to use those powers.
Second, many campaigns continue on past level 20. In previous editions there were books published for just that purpose, and one can easily assume those will come for 5E if it is successful enough. Even without them, one can continue to level following the standard progressions, and potentially house-ruling new, cool capstone powers for L21+. I'd be willing to bet if you look you can probably already find some of those on the internet.
I also just learned that the 5E DMG (which I do not own) already has suggestions for continuing past level 20. Instead of continuing to level, players get Epic Boons (a.k.a. super Feats) at certain XP increments.
Third, your question is self defeating. If there was no power at level twenty, people would either ask why not, or complain that there was no point to level 20, and say that level 19 was the effective cap. At which point the same thing can apply to level 19. Repeat ad nauseam.
First of all, as has been mentioned, campaigns do sometimes see actual play occurring at 20th level, or even higher. So your question actually includes some assumptions that are not quite true.
However, I do some third-party work for D&D 3.5 and its spin-off Pathfinder, which, like 5e, have a cap at 20th, and in practice, are rarely played at such high levels (most campaigns actually stay at quite low levels). Nonetheless, “capstones” are still quite important to a class’s design, even if they are rarely used.
So this is an answer to your question, accepting your premise that 20th-level class features will be rarely used, if ever. Again, note that this is not necessarily the case.
Capstones are still important because they reflect the ultimate power of that class. They represent the culmination of all their training and skills, and what they can now do. And as you say, they’re awesome—and as a designer, that’s the most important thing in a capstone to me. The capstone is supposed to inspire you, excite you. Get you to really love the class’s potential.
They’re also an opportunity to flex a little as a designer. Like you say, a campaign is probably nearly over by the time you get one, if you ever do. The capstone represents an opportunity for the gloves to come off and to write something really crazy. That’s not to say that there are no rules, or that nothing is “too much” for a capstone, but you can “get away” with a lot of things that would elicit a negative reaction from readers if offered earlier.
The DMG has continuing increases in power for 20th level characters. The capstones are the reward for being single-classed through to "epic tier"... but the Epic Boons are quite nice, too.
It's thus clear from the DMG that the intent is to allow continued play at 20th level. Note, no provision for higher levels, just additional feat-like boons at regular intervals of earned experience.
Note that while characters cannot exceed level 20, level 20 isn't the end of power growth, just the end of HD, HP, and Proficiency Bonus growth.
Capstone powers/abilities/whatever you want to refer to them as (generally just class features) are the penultimate potential of the class. Getting them says that you are a MASTER in that class. It's the reward for sticking to the class to the bitter end. The purpose is to use them in the final dungeon. You are now a legend. You are a kung fu master, a black belt in karate, a super genius archwizard, the archbishop of your church, or whatever. (Not strictly speaking those exact roles - some require you to be ordained or promoted in station, or whatever, but that illustrates the point).
These sorts of things are epic by definition, though the game has special rules for "epic play" - beyond 20th level. At that point you're effectively a minor deity in your own right. At such a point, there's usually nothing left to do than literally save the world from the all-powerful Lich King, the Final Wyrm, the Destoryer of Worlds. You can now go toe-to-toe with the final boss, the whole point of most campaigns. Your challenge will take all of it out of you, but 20th-level powers allow you to succeed against 20th-level baddies, which is generally what the Big Bad will be (or higher! And he/she/it should be even more challenging with cunning plans, powerful spells, mighty attacks, an army of minions, a temple or dungeon of pain, etc. Might even have a god on their side!)
The Magical McGuffin - some kind of artifact that will save the day from the villain - will be used.
All the world holds its breath; the outcome of this final battle will decided the fate of the very world! It's Good and Evil's final battle, and these abilities facilitate combat on the tier reserved for only the masters.
Some campaigns can and do play at and go beyond level 20, so capstone features do get use for those games.
But another thing to keep in mind are magical items like the Deck Of Many Things that can boost your experience or level. So even if your campaign is over or virtually over at level 20, it's possible one or more of the players can get to 20 early through use of magical items, and that point can benefit from their capstone as a reward.