Because at least you put some thought was put into it, which again, is better than no thought at all.

Consider your question applied to math: how is a false mathematical argument for believing a theorem better than just believing it for no reason at all? It's clearly better because you can actually point out the flaw in the specific argument.

If someone accepts a mental framework and is faced with a choice they want to make, either:

1. They can find no justification for a choice they want to make, at which point they are more likely to question whether their choice is justifiable, or their mental framework is sufficient. Either outcome should be encouraged.

2. If they find a justification that's valid in the framework, then they have an explicable basis from which to convince others they made the right choice.

3. If they find a justification that's fallacious, then pointing out either how the framework is incomplete, or their argument within the framework is incorrect is far more likely to change their minds than simply claiming their choice was wrong and trying to explain why from your own framework, which they haven't accepted (and likely won't without a lot more convincing).

Ultimately, applying thought to a problem is always better than no thought.

I think admitting a choice is arbitrary will often require more thought than a specious conclusion arrived at by a faulty process.

Doubtful. Like the OP said, "because I want to" is all the thought it requires. It's the answer children instinctively give. How much thought do they give their justifications?

Finally, if you're giving credence to a process, even a faulty one, then at least it's possible to erode confidence in the process or your application of it. To see its effectiveness, you need only browse the countless posts on reddit written by former Christians about how they came to question and ultimately give up their religion after a discussion with an atheist.