A few reasons:
In the States at least, many of the garage doors on the market now don't use regular contact closures to trigger their up/down motions, but instead some sort of communicating protocol.
This means that there is no direct way to ask it to go up and down in the absence of their controllers - hence my statement about modifying an official controller.
You could indeed go through the effort to modify the opener control circuit itself, but this might be much more difficult than you imagine. Modern openers are quite technically complex and "hacking" them is much easier said than done. There won't be simple up/down controls on them, as a micro controller will be running everything, a micro-controller which receives commands over it's comm bus. You could hack the comm bus, but I doubt it'd be worth it.
As to running the motor directly, three problems. First: A meaningful percent of openers are using motors which need specific motor controllers - controllers you would need to cut out if you're going to try and pull out it's control solution. This isn't a problem on all openers, but it is on some of them. Second: If you do the modification, no garage door repair company will ever touch it, so whoever does the modification will be responsible for keeping it working forever. Third: There are actually a lot of building codes and UL listing guidelines around openers. You'll be breaking almost all of them to perform this modification. In doing so you inherent a lot of liability. Hopefully nothing bad ever happens, but if it does you'll be liable for it. For all these reasons I wouldn't do it.
But all of this is besides the point. I answered the question presuming that the poster had a pre-existing consumer door he wanted to control. Thus I answered the question of how that is typically done, in the States.
If you wanted a more robust answer to this question, you'd just buy an opener with good support for remote control systems. They exist and are relatively common here.
An opener such as this is not only more robust than your typical opener, but will also easily work with an automation system while maintaining all its listing and code needs.
Thanks, Karl P