The Euclidean algorithm finds gcd(84, 30) by factorising both numbers and comparing their prime powers.

False

False. It takes repeated remainders — $84 = 2(30) + 24$, then $30 = 1(24) + 6$, then $24 = 4(6)$, giving 6 — and never factorises anything. That is exactly why it is preferred: remainders take logarithmically many steps, while factoring a large number has no known efficient method, an asymmetry RSA depends on.