Clearly we are talking about wildly different scenarios, and that is causing considerable confusion. For examples:
> I severely miscalculated and waited too long to make camp after it became clear that
> I couldn't dodge the storm.
and
> If we're hiking in the mountains, we are paying attention to the sky, especially in the afternoon
Australian and New Zealand walkers are used to dealing with bad weather which lasts for the day (or more). The idea of a brief storm which can be dodged is … amusing to us. The idea that storms usually arrive in the late afternoon – forget it. But I understand that these ideas are relevant for some parts of America. Different places, different conditions.
> multi-day storm … Normally, I'd just hole up somewhere (tent or town) and let it pass.
Huh? For a start, we (Au/NZ) don't have towns in our mountains where you can 'hole up'. Secondly, the idea of holing up is not one we normally even consider. We just keep going. Bail-out points may not exist, and we have to get to the other end on time. Different places, different conditions.
> In terms of setting up double-wall tents that aren't integrated whilst in a
> pouring rain, well, if you've practiced at all the inner might be exposed to
> the torrents for 30 seconds.
That *might* work if there is zero wind. But what if the storm includes a 40 mph gusting wind? I think you might find it takes a lot longer than 30 seconds in many cases to get a loose fly over a pop-up interior. In addition, you might need several people to just hold the pop-up interior upright in the wind – it won't have any guy ropes on it. And then, when you have the fly thrown over the poles, you will need several people to hold the lot up while you get the guy ropes staked out. Even so, when the fly is not strapped to the poles the lot can collapse.
There have been some really graphic videos on YouTube about this recently – very illustrative.
I am not saying anyone is wrong here about handling their own conditions. I am sure that what works for you is fine – for you.
But I am saying that extrapolating from conditions you are familiar with in your locale to conditions far away or even in a different country is risky – sometimes even foolish.

The top photo was the late afternoon. Lovely weather.
The bottom photo was the next morning. The storm arrived in the night and lasted through the next day, with winds to 50+ mph.
Not quite the same as a fine night in the pine trees …
Cheers