Just because a task is "done" does not necessarily mean you have met the original requirement. Sometimes you might not meet all the requirements, but the tasks are still done done - eg you had a task to run a test and it failed to meet the spec but maybe you agreed with the customer that this was acceptable in the end. The RTM looks more at the output of tasks, rather than if the task is done or not. Individual high level requirements also might be spread over a number of tasks if they involve different teams/technologies/subsystems so it is not always the case that one task = one requirement.
I don't think it is a good idea to use your schedule as a RTM. The RTM is a 'one stop shop' giving yourself and the customer visibility of the overall requirements traceability.