My guess is that the Pope would tell the person to join whichever Sui iuris Church (of whatever rite) that led said person to the Catholic Church- whether this be the Roman Church, the Melkite Greek Catholic Church, the Ethiopian Catholic Church, etc.
I’m not sure why you responded with “The Church is not a carpet”. The point of referring to the Church as a tapestry is to show all the wonderful liturgical and devotional traditions that She contains. Also, in respect to relationship between the Roman Church and Her sister Churches, there is no division as you state. They are a seamless whole; though they be woven of different colors and clothes, they form the seamless Coat of Many Colors, the very own robe of Christ that He has placed upon His people.
You are correct in stating that the Church is a teaching institution, insofar as one of Her charged duties by Christ is to teach. But that is not the exclusive duty of the Church. The Bride of Christ has many facets, and teaching is but one of them. She is to also exercise mercy, care for the poor and needy, tend to the sick, offer up worship, and be the Mystical Body of Jesus upon Earth. She unites Heaven and Earth together for She is like Eve, formed from the body of Her husband; and Her Bridegroom, Jesus, unites Heaven and Earth by His salvific Incarnation, Passion, and Resurrection. Jesus unites Divinity and Humanity, and the two twill never separate.
I would implore you, sedonaman, to investigate the Eastern Catholic Churches. On a larger scale, they are the sister Churches to the Roman Church. On a smaller (and yet equally important level), the members of said Churches are your brethren in Christ. You owe them filial love and respect, and part of that is understanding them. Whether or not you speak in ignorance or malcontent, I know not. It doesn’t matter. What does matter is that such talk be corrected.