Are my prayers accepted despite my not saying the opening Takbir (Takbirat al-Ihram) in my youth due to ignorance of it, and does the Hadith: "Raised from my Ummah are error and forgetfulness" apply to me?

The opening takbir (Allahu Akbar) is a pillar of prayer. There is no need to repeat past prayers performed without the opening takbir if the person was unaware of its obligation. This is the preponderant opinion among scholars, because the Prophet (peace be upon him) did not order the one who prayed incorrectly to repeat past prayers in which he omitted ṭuma’nīnah (tranquility) due to his ignorance; rather, he only ordered him to repeat the current prayer. Nevertheless, the person is sinful for his negligence in learning the rulings of prayer, as ignorance of them is not excused.