Well, it asserts it as a right "when a long train of abuses and usurpations, pursuing invariably the same Object evinces a design to reduce them under absolute Despotism"
Which is not that different than "all legal avenues exhausted" but you're correct, it does claim it as a literal right. Thanks for inspiring me to re-read it.
If the DoI is a reflection of understood common law, and it asserts a right to rebel (which, of course, it does), then there is a right to rebel.