1. Itachi can't use Izanagi because he doesn't have Senju DNA, it's as simple as that
2. His mentor, was a liar and deceived people, couldn't that also imply that he kept the truth from Itachi as well?
3. You're right, even though it wasn't mentioned that Madara didn't have access to Senju DNA, but I won't make any assumptions that was not stated in the manga
4. "It is heavily implied" that doesn't make it true. Once again, Senju existed back in that time. Until you post a manga scan of an Uchiha using Izanagi without Senju DNA, my point stands until otherwise. It hasn't been done and won't be done but the conditions have been made clear