Having bought and sold a few houses in the past decade, I think the most important thing for us was picking a lender with a local branch (we used a local branch of a major multinational - TD Bank - as they had the best rates in our area, even when comparing online).
Once, when things were getting close to closing, the "corporate" office of the bank (loan originator) was not doing its part to get everything done and we were only able to close on time by having the local branch manager, our RE agent and lawyer all work together to get it done. The manager got us a significant extension at no cost - which helped save the deal (Interest rates at that time had gone up significantly since we locked at it seemed like the corporate office was doing everything it could not to close at that rate)
If it was just a refinance, I'd say the best rate regardless is fine, since you are in the house and don't have a timeline with a seller. But buying, especially if you are in a "hot market" and with rising rates, I want someone to talk with, face to face, to help out if needed.