Does this proof sketch show that finding a SHA-1 preimage is NP-hard?
they provide a proof sketch (Appendix A) to show that finding a preimage for SHA-1 is NP-hard


No, they don't.  The definition of a problem being NP-hard is "if you have an Oracle that can solve instances of this problem in polynomial time, you can solve any problem in NP in polynomial time"; as we can quickly find SHA-1 preimages using a huge lookup table, we can easily see that finding SHA-1 preimages is not NP-hard (unless P=NP, in which case every problem is NP-hard)

What they do (at best) is show is a class of problems that are jointly NP-hard (of which finding SHA-1 preimages may be an instance).  However, that doesn't tell us anything about the concrete difficulty of SHA-1; NP-hard problems can have easy instances (trivial example: travelling salesman problem over every city in the world; is there a circuit that's at most 1 kilometer total distance?  Even though the traveling salesman problem is NP-hard, this is a trivially solved instance)