No it should not lock. I own several that do not lock. Been common on GM and others manufacturers for years now. Really has nothing to do with push to start either or having a keyed ignition. My '08 Civic is keyed ignition, steering does not lock, '11 GMC Sierra SLT, keyed, does not lock, '13 Cruze push to start, does not lock, and '15 Impala LTZ, push to start, does not lock.