Why pay for something you don't need?
Traditional desktop computers are slowly but surely being phased out. The days when every household had a cheaply made (but hardly inexpensive) Compaq or HP computer and clunky monitor for the family room are long past, and outside of people with extreme needs only businesses are buying desktops. And not because our needs have drastically changed, but because laptops, convertibles and micro PCs can do all the things we want them to do. And it's great not being chained to one desk to work and play on "the computer."
First ask yourself if you need a desktop at all.
Where does a Chromebox fit here? That's a question plenty of us have, including me. A Chromebox isn't something you can just recommend to anyone because for most people a Chromebook or something like a Pixel C or iPad is just better and not stuck at home. Companies making PC hardware know this, too, and you can buy a surprisingly good laptop for around $300 now from Google or Microsoft. But there are some cases where a desktop computer is practical or even better suited than a fancy thin flippy book thing or a silver laptop.
We can start with a case where a Chromebox just isn't going to cut it: gaming. Native Client and even HTML 5 were/are capable of some impressive stuff. But Chrome is caught in the catch-22 of nobody making hardware that's really good at playing games because nobody is making really good games. Other than a handful of "adult" game clients (those ads you may have seen daring you to try at a popular adult video site) the only really cool "games" ever developed for Chrome came from Google as API demos. The landscape is bleaker when you want to add VR to the mix.
We're OK with this. While it would be nice to have a few showcase games for Chrome, we don't want it to turn into something that isn't cheap and easy, like the money pit and time sink many of us have to play all those AAA titles. If you need a desktop computer for gaming, don't waste your time looking at a Chromebox.
You don't want a Chromebox if you're a gamer. Use the right tool for the job.
But there are two other areas where a desktop PC could be a better choice than something portable, and chances are a Chromebox (or a Chromebase) is the best way to do them both: something for the kids or something for the entertainment center.
For the kids
Working mainstream technology into your kids' lives isn't easy. The products designed for children are safe and fun but can be a little boring once kids get to a certain age. And deciding when and how to introduce the young'ns to the internet and all the horrible things that live in it is one of the hardest things a parent will ever do. You can make the hardware part of the decision easy and buy a Chromebox. Chrome has a built-in supervised account feature, but there are also many other options available for parental controls through cloud services, application monitoring, and browsing proxies. I wish I had these things available on a sub-$200 computer when my kids were growing up.
Older kids can benefit from having a Chromebox if their school uses Google Apps for Education, too. The school login can live right beside a home login with different permissions for each account.
For the home theater
A Chromebox connected to a television is a no-fuss cheap way to make any model smart. A flip of the input source on your remote can take you to a full Chrome OS experience, complete with the best web browser available and all the apps and extensions you already use. It's the perfect gateway to all your online entertainment as well as a front end for your local video library through a NAS or media server. Best of all, you already know exactly how to use it.
Chrome is the best way to find the best media the internet has to offer.
Android TV is great. I love my Shield TV and the world of apps and games and everything Google has to offer through it. But sometimes a web browser is just the best way to do things, and for entertainment, this is the case. Netflix.com is a thing and makes any Netflix app unnecessary. And why buy a traditional desktop PC for hundreds more when the first thing you'll do is install Chrome?
If you have particular needs that require a Windows or Mac desktop PC, you know this and know that a Chromebox can't replace it. But some things a desktop does better than a laptop are also better done with Chrome.