Are the steps mentioned correct, given that a person from Egypt arrived in Riyadh with his family, and after an overnight stay in Riyadh, they headed to Mecca to perform Umrah in Ramadan from the Miqat of Sayl, and remained in Mecca until the months of Hajj, then entered into Ihram for Umrat Al-Tamattu' leading to Hajj from outside the boundaries of the Haram, and went out to visit the Prophet's Mosque once before returning to Mecca to await Hajj?

What you did is correct, but your travel to Medina after performing Umrat al-Tamattu‘ might interrupt it, due to the اختلاف الفقهاء (difference of opinion among jurists) concerning one who performed Umrah during the months of Hajj then traveled a distance permitting Qasr (shortening prayers) before the Day of Tarwiyah: whether their Tamattu‘ is interrupted, necessitating Ihram upon their return to Mecca, or not. You may adopt the opinion of those who hold that Tamattu‘ remains until the Day of Tarwiyah, so you would assume Ihram for Hajj when you enter Mecca without needing Ihram upon your return, even if it is not the preponderant view, as it is permissible to act upon a less preponderant opinion after an action has occurred and it is difficult to rectify.