As it stands right now, an HTPC is going to do the most separate functions, but it's not necessarily the best option for all services. Here's my cord-cutter setup, and it works great for me:
Roku for Amazon, Hulu+, Netflix: Clean interface, super low-power, remote easy enough for my daughter to learn at 4. You also get the 300+ other channels for Roku, plus a bunch of awesome private channels (and the ability to create your own, if you're so inclined). It's NOT a local streaming machine, though it's possible via a variety of hacks. Overall, we use this the most.
HTPC for DVR functions and various internet stuff: I like Windows Media Center, but there are a bunch of different free flavors out there, depending on what hardware and software you're using. It's a great DVR for our OTA signal, plays every format available for downloaded stuff, and handles DVD rips by default. It's also the best way to play the stuff that content providers won't allow on Roku-style set-top boxes, like standard Hulu. Oh, and it does music, too, of course.
TV Tuner for OTA: I use my HTPC for this sometimes, but really like the speed and ease-of-use of using the built-in TV tuner. YMMV.
The main point here is that no one device is necessarily going to do everything you want it to, or not in the way you want it. A HTPC is the most versatile, but not always the prettiest or easiest for everyone in the family (if you're techie but your spouse/kids aren't, for example). I love my Rokus, and they're so cheap that it wasn't a big deal to get one for every TV. I don't miss cable at all.