Is the aforementioned method for performing Salat al-Istikhara—which includes reciting specific verses and supplications, then opening the Mushaf and seeking optimism by looking at certain letters after the word of majesty "Allah" on a specific page—correct and legitimate in Islamic law?

The legitimate Istikhara is what was mentioned in Fatwa no. 43087. The method mentioned in the question has no basis. Allah accepts from deeds what conforms to His law. What has been authentically reported from the Prophet, peace and blessings be upon him, is sufficient for anyone who wishes to perform Istikhara. Indeed, good lies in following [the Sunnah], and evil lies in innovation. The ruling on seeking good omens from the Mus'haf has been clarified in Fatwa no. 46037.