No. There is no way for Bitcoin mining to "break SHA-256 and render it insecure". There isn't enough energy in the whole SUN to do that.
1. exploiting cryptanalytic weaknesses in SHA-256,
2. successes in new computing paradigms such as a large scale quantum computer using Grover's algorithm.
3. or turning the local galactic cluster into computronium for SHA-256 miners. Although at that point we have bigger problems. =)
If you think SHA-256 will not be broken anytime soon then that suggests that you believe that mining difficulty increases will slow down.