I can't confirm his reasoning from any specific quotation from the book, but it is very likely that Snape believed he could break the curse. He would have no motivation to break it for other teachers, but had he held the position himself, he would have likely done everything he could to break the curse. The curse would have already been a decade old at any point in Harry's Hogwarts career, plus Snape had access to libraries and continued learning in order to research how to break it.
It is not unreasonable to believe that Snape had powers the Dark Lord didn't know of. For instance, Snape was the superior legilimens and Voldemort never understood Snape's motivation.
Also, to some extent, Snape broke Voldemort's curse on the ring horcrux/hallow. He told Dumbledore that he could have done more had Dumbledore a.) Called him sooner, and b.) Not actually worn the ring.
Snape's Dark Art powers definitely rival Voldemort's.