Is it important to do a PhD at a different institution than undergrad?
He used a lot of hyperbole which has made the decision seem dramatic.


We do not know this person, but it is possible that this is the professor's way of saying he wants you to change your plan for a reason he does not want to tell you.  Perhaps he thinks the PhD supervisor you are considering at your own institution is not suitable, and he does not want that professor to realise he thinks they are unsuitable.  So he is telling you indirectly.  

Reflect on past advice this professor has given you.  Was it reliable?  Was it coded?  Ask more faculty for advice.  What do they tell you?  Ask your potential supervisors' former students what they think about the supervision they received.