1. Will I trust this person to do their job competently?
2. Would I trust this person not to defect in a prisoners dilemma situation?
The second is what you need the personal connection for. There are two ways to work on a project, to make sure it has the highest chance of success, or to make sure when it fails you don't look like your to blame.
#2 is needed so that both parties take a chance on each other and don't optimize for throwing the other under the bus.
People should be able to trust their coworkers on ways that the GP didn't capture. But those do not include adversarial situations. (And honestly, "Hey, how was your weekend?" can as easily add or remove all kinds of trust. None of this is simple.)
Many of the managers I've worked with spend as much if not more time thinking about CYA/metrics/etc... than accomplishing the tasks they're given. And the ones that don't spend a lot of time developing relationships instead.
Do you work for the mob?
When those products/projects inevitably don't deliver on some expectation, someone is going to ask why? It is pretty important when that happens the answer isn't "jamesbarney is an idiot".
Here's a subtle betrayal that can happen, it comes raise time, you've been busting your ass on something that is really beneficial for the project but doesn't help any of your KPIs. Is your boss going to go to bat for you pr are they just going to tell you how important your KPIs are and wish you better the luck next year.
This is bullshit. I trust my accountant because he's competent and honest, not because he's been vulnerable in my presence. What you're describing is unprofessional.
The last thing I want to know about (or experience from) my co-workers is anything personal or intimate, unless it's relevant to the job.
If I want to know my coworker better that's my choice, but otherwise it's somewhere between irrelevant and unwanted.