The real question is not online versus local, it is speed and responsiveness once you have an accepted offer. A lender who can turn around a property-specific approval within your financing condition window, typically 3 to 5 business days, is more valuable than one with a lower advertised rate but a slower process.
Your agent should not push you toward a specific lender for a referral fee. A reasonable ask is that you get a second quote from a local option so you can compare speed and terms, not that you abandon a pre-approval that is already in place.
The goal is a lender who can actually perform on your timeline, wherever they are based. Get your hand-reviewed valuation from Joel Dyck.