Is my prayer for missed prayers valid in the manner mentioned, by limiting it to Al-Fatiha and a single Tasbeeh in bowing and prostration, and omitting the remembrances, the regular Sunnah prayers, and the Witr and Shafa' prayers?

Praise be to Allah for your repentance. There is a difference of opinion regarding making up prayers intentionally missed. The safer course of action is to make them up, and the method of making up prayers is explained in Fatwa no. 72777. If the missed prayers are numerous, it is not prescribed to make up their Sunnah (supererogatory) prayers, except for the Sunnah of Fajr, and you have a choice regarding Witr. You are permitted to confine yourself in the made-up prayer to only the pillars and obligatory acts, and reciting Al-Fatihah and the glorifications of bowing and prostration once is sufficient. "Rabbighfir li" (O Lord, forgive me) between the two prostrations is obligatory according to the Hanbalis, so we advise you to perform it to avoid the difference of opinion. Your past prayers are valid, Allah willing, due to your ignorance of this matter.