I would like my users to be able to click a link straight to a new task form, then submit, but never see the actual task list. I want to be the only one that can ever view the task list and especially edit.
Being able to submit also means ability to change/update = Contributor (at least). So "hiding" in your case would probably mean you overriding a view with the UX your wish, e.g. via JSLink or better Client-Side rendering on a Custom task list and handle updates behind the scenes. That would of course, eliminate the need for people to use the standard UI, but it won't really, "hide" the list from knowledgeable people.
Now a 2nd alternative would be to rely on a Full farm solution and use Elevated privileges (https://msdn.microsoft.com/en-us/library/office/microsoft.sharepoint.spsecurity.runwithelevatedprivileges.aspx) to go around the permissions needs in your custom web part. The issue would be, of course, that you would be loosing the "Modified By", etc. as you would be executing your code with Site Collection administrator permissions, not to mention that you'd have to deploy binary code (e.g. WSP solution package) on your farm.