They thought wrong as Christ made it clear therefore we know the difference.
"They also believed that their Tradition was handed down from God to them alone. Yet they were obviously wrong."
I'll need the source to reply completely. They certainly had authority as Christ indicated in Mt: 23:3.
But their very Tradition blinded them to Christ. They could not accept His authority.
For the source, look into the Talmud on who and what the Messiah was supposed to look like. Forgive me for not having a link handy, as we are just about to run out the door.
Also, I am not trying to throw bombs (even though that is a weakness of mine) but trying to point out something. Traditions can blind you just as much as they can protect you.