What advantages do "alt-coins" have over bitcoin? I'm struggling to see the importance of alt-coins. It seems to me that one of the major selling points of the various alt-coins is their quicker confirmation times. The common argument is: this is good for merchants and the like, because there is less time for them to worry that the customers' payments are legitimate.

But can bitcoin users not have the same "luxury?" Why can't a merchant run his own node? His node will process the transaction, and the rest of the network can play catch-up.

Another argument is that there are too few bitcoin, and that eventually bitcoin will be the gold of digital currency--what the big transactions happen in--and the alt-coins will be the silver, or the copper--for day-to-day use.

Is that really even good in theory though? The price of bitcoin would need to be in the millions for that to even be a decent worry. I can possibly see it being an issue, but do we really need to consider it now, when the exchange rate for bitcoin hasn't seen more than a few months in the 4-digits?

Am I overlooking a hyper-important advantage?
It depends a lot on the specific altcoin you're talking about. For the most part, there aren't a lot of major differences between Bitcoin and the alts. Sadly many of them exist only as the leftovers of pump-and-dump schemes intended to make their creators rich, though there are a few that are interesting in important ways.

One of the primary benefits of a thriving altcoin economy is that they often represent the testing of a theory that may eventually benefit Bitcoin itself. Many altcoins, for example, use scrypt as their mining algorithm instead of SHA256. If SHA256 is discovered to be compromised some day, having a drop-in replacement for that subsystem available would be a major boon.

You specifically mention faster confirmation times as a potential benefit, but the most often overlooked part of that statement is that transaction finality is a result of computational cost, not of time. This means that if an altcoin has one tenth the confirmation time and an equal amount of hashpower relative to Bitcoin it will take ten times as many confirmations to reach the same level of transaction security. Faster confirmations are a psychological benefit, not a technical one.

The "silver to Bitcoin's gold" theory, largely pushed by Litecoin proponents, is also fallacious in that the entire reason gold requires a baser partner metal is that it is not adequately divisible. Paying for a candy bar with gold would be incredibly difficult for both parties involved since the amount of gold needed would be absurdly small. Bitcoin is not subject to such physical limitations and as such can be divided into such tiny portions relatively easily.

Taking all of the above into account, it should be clear that while a handful noteworthy exceptions do exist, the majority of altcoins are intellectual curiosities at best and scams at worst. A few make meaningful changes, but thus far none of those changes has really been a meaningfully large improvement over the base system.