How many times do I need to go down this path:

  ...
  - ask for help
  - try to explain how it's supposed to work
  - try to explain how it's not working
  - Listen to them come up with all the ideas that I've already ruled out
  - Try to explain why their preferred idea has already been
    ruled out or is not actually relevant to the problem
  - conclude they still don't understand the problem or even
    how this part of the system works
  - go ahead and just solve the problem myself
  ...
  

before it's OK to stop "asking for help", and just fix the damn thing?

I could see this the other way, where the reason this path happens is because of a problem with how the team operates so others are not in the loop on the problem or , even worse, the entire system. I think what the parent is proposing is trying to combat that entire issue

Possibly in an ideal environment. But expecting the same someone to work with me for an entire week to either debug or design usually hasn't been realistic. Keep talking with different people, or the same person on and off has a huge context cost.

What I've found to be a good middle ground is take some time to think things through myself first, at least in rough strokes. Then have someone else come in and review the work. There is a balance of doing too much vs too little yourself of course, but I don't think there is a hard and fast rule (e.g. 1 day worth) for it.

I'm confused about why you're suggesting the only options are "work together for an entire week" or "don't keep anyone updated about what you're working on for a week".

There are a lot of ways to keep your team updated, but more importantly, it's frankly arrogant to think you can or even should tackle any problem on your own.

Joe is the expert on feature X. When we have a problem in X we ask Joe for help and he invariably provides the insight needed to solve the problem.

When Joe has a problem with X, he should go ask for help, even though he's the one best able to deal with it? Should he ask himself for help?

Can you see the problem with blanket statements like this?

I've been in this situation many times too, but I have to say it feels like a weakness. When a single person works on a problem I see improvements being left on the table compared to when two people effectively collaborate. I've experienced this in a wide range of skill/experience levels so I don't think that is the problem.

That said, I don't know how to change the situation if you have devs with skillsets that don't seem to overlap much, which seems inevitable for some companies

In general I agree, where possible and where it makes sense getting others involved early is a good thing. Somethings are more solo work however, at least for a while to get things oriented. Some times it takes time to gather enough context to even start to express a problem space to others. Not everything is just adding a button to a web page.