that doesn't make sense though, since applying a label and then archiving the email, or doing it in one step by using the "move to" button, is the EXACT same as moving the email into a folder, except it isn't physically located in some folder. A folder is essentially a completely inflexible label. You are saying that an email belongs to this label and this label only when you put it in a folder. Using labels instead of folders allows you to have ALL the functionality of folders, but with the ADDED functionality that an email can exist in multiple folders/labels at the same time.
If you want to use your inbox as a to-do list, well, leave emails in your inbox until the item is complete, and then when it is complete, archive it. Labels actually HELP in this regard, because you can label something so that its context is easily identifiable, but it can still exist in your inbox until it is complete, at which point you archive it. With folders, you can't sort in this same way while things are in your inbox. You have to move it out of your inbox in order to sort it into a folder so that it has context, such as moving it into a project folder. I can label my emails as being associated with a particular project, and keeping it focused in the inbox as an item that needs to be taken care of using the label method. This is not possible with folders.
Your list of labels is on the left, just like a list of folders would be in any other mail program. When you click on a label title it gives you all the emails with that label, which is the exact same as clicking on a folder name and seeing all the email that is located in that folder.
You haven't demonstrated any functionality that folders have and labels lack.