Under California law, the implied warranty
of habitability is, in effect, a corollary to a residential landlord's statutory obligation to put the premises “into a condition fit for such [residential] occupation, and repair all subsequent dilapidations thereof, which render it untenantable.” Civil Code
A defective garage door does not render a rental unit untentable. It's a defect, for sure, and it would entitle a tenant to use the "repair and deduct" remedy of Civil Code 1942, after 30-day notice of the defective condition. But, the defective garage door will not operate as an affirmative defense
to legal action for a breach of the lease
-- nor, will it operate as grounds for constructive eviction.
Your recourse is to show that the landlord has made no good faith efforts to relet the property after you vacated the premises. A landlord cannot collect for unpaid rent, unless the landlord mitigates damages by making good faith efforts to find a new tenant.
I realize that this answer may not align with your needs, but if you want a positive legal outcome to this issue, you need to know exactly where you stand.
Please let me know if I can be of further assistance.