According to David, who alone were to carry the ark of God?

Answer: The Levites

After Uzzah's death David said no one but the Levites, chosen by the LORD, may carry the ark.

“Then David said, “No one but the Levites may carry the ark of God, because the LORD has chosen them to carry the ark of the LORD and to minister before Him forever.””— 1 Chronicles 15:2 (BSB)