If I'm going to have surgery done, the single most important factor in deciding who my surgeon is (besides what insurance I have), is whether or not I trust they can get the job done right.
I don't care how terse they are in the consultation. I don't care that they didn't ask about my hobbies.
Trust is all-important.
But that's not the end of the story.
High School Debate
I took debate as an elective in high school a whopping 6 times, and participated in tournaments enough that I earned a membership in the National Forensics League (NFL).
NFL debate tournaments were hosted at a variety of surrounding high schools (complete with trophy ceremonies, and everything). This is where I learned how to be comfortable speaking in front of a crowd.
The set-up is like this: There's a "should" statement called the Resolution. For example, "We should respect the right to life." Then you're assigned the Affirmative role (you support the resolution), or the Opposition role (you argue against the resolution).
The Affirmative debater would go first to build their case. Then there's 3 minutes for cross examination where the Opposition could ask clarifying questions. The Opposition could choose to go with the straight refutation strategy (only focus on why the Affirmative's case doesn't hold water), or the Refute & Build strategy (argue against Affirmative, and then present their own case to go against the resolution). Then Affirmative has a chance to speak again before the Opposition has the closing remarks.
That's my whole world from 9th grade on through into college.
What does all this have to do with liking & trusting?
Everything, really. The question of whether trust or likability is better is a perfect example of a beginner's debate tactic:
The false dichotomy: A logical fallacy that artificially assumes there are only 2 outcomes when there are, in fact, many other possibilities. Often called black-and-white thinking.
The question "which is more important" tends to direct your thinking into focusing solely on one or the other, instead of realizing you'll do a lot more business with both.
Turns out, doctors who spend a little more time with patients, being personable, paying attention (instead of only looking at their charts), etc. are sued less often than doctors who are more to-the-point.
It's in everyone's best interest is the doctor is, in fact, trustworthy and likable.
Same goes for you.
Sure, I can do business with someone I don't like if I trust they'll deliver what they promise, but I'm going to continue doing business with someone I like and trust.
Being one doesn't preclude your ability to be the other. It's not a binary system. Being likable and trustworthy will gain you business, as well as prevent you from losing business in the long run.