One can fully torify a system using Qubes OS TorVM, Whonix or a custom transparent proxy. Everything gets torified, even Adobe Flash.
Why should one use Tor Browser even though everything is already torified and can't just use Firefox or Chrome?
Using such a system only prevents IP address leaks. The Tor Browser has many, many patches to reduce fingerprintability and fix privacy-breaking bugs. Think of it like this: Does it really help that no one can get your IP, if they can potentially link up all of your online identities? There are various and subtle ways of tracking a browser across the web that popular anti-tracking addons do not defend against; without the advanced work of the TBB there is a chance you'll end up in tracking databases anyway.
Configuring another browser to connect using Tor is reasonably easy. The more difficult problems are:
The Tor Browser is equipped and configured to make it very difficult to tell one Tor Browser user from another, but it doesn't hide the fact that you are using the Tor Browser. Torrifying your own browser can make it more difficult to identify you as a Tor user. That said, if you overlook anything, make a mistake with your setup, or fail to secure it against undesired changes, you're uniquely identifiable. Most users don't have the skills necessary to accomplish these things or the patience to examine and test all of the necessary details. It's entirely possible to make another browser work properly through Tor. It's just a lot more work than most realize.
It's pseudonymous rather than anonymous.
What's the difference between pseudonymity and anonymity? That deserves its own question.