These missiles are not especially fast given the distance they have to cover. They were in the air for 10+ minutes, more than enough time to cross check IR early warning and radar tracks.
I should clarify that it is ENTIRELY possible that you are correct. It is entirely possible the US made a choice to not intercept here. My point is that it is also entirely possible that the US didn’t make a choice. I just don’t see enough clear data to tell. I also think that, even if it was a choice, it was probably more of a split-second decision made by lower-level officers on the ground as it happened, rather than a carefully-considered decision made by national security leadership.
The chain of actions you're describing would have around 10-15 minutes to be executed.
The likelihood that neither THAAD nor any of the Patriot batteries were sufficiently ready to launch is also basically zero.
THAAD has an engagement range of over 100km, so the battery was undoubtedly in range; and it can cue Qatar's own Patriot batteries so they can launch the missiles; I find it hard to believe there were none in Doha.
So, this would not be a split-second decision - there was quite a bit of time given the proposed trajectory. It's very very unlikely the US didn't have the sensors to see the missiles given the intense concentration of long-range ABM radars in the region.
Is it theoretically possible the US legitimately couldn't do anything? Yes. But the odds are overwhelming that the US knew about the strike with minutes to spare before making a decision, and decided not to protect Qatar.