Why can you only assign a quick task to your inbox or an existing project? Surely you should be able to add it to "Today"?
First off, every tasks needs a project and that's why you need to select a project beforehand. When assigning a quick task without a project, it would have to go somewhere otherwise you wouldn't be able to access it anymore. Say you can put it in "Quickly added tasks or For Project Assignment" so you can access them, it would be possible for it not be a project though if you group them, it would be just the same as projects. Also if you just put it to "Today", when you remove the date, where will it go to?
The Inbox works great for this as it's the place that you can dump tasks that you can't process at the moment. You can then review them later and assign them a project, date or any other details that you need to add.
Finally, feel free to add this a suggestion in the Votebox https://todoist.com/Vote/. You can include details on how you want this work and other users can vote for it.