The US probably was able to detect the launch, yes. But would they have assets in place to conduct an intercept in time? Different systems are meant for different targets, and physics put fairly stringent requirements on where an interceptor launcher has to be to protect a specific area. Plus you've got only a few minutes from missile launch to impact. A few minutes to figure out missiles have been launched, confirm it's not a bug or an anomaly, figure out where the target is, get that information from the person monitoring the sensors to command and control plus the missile launch crews, get authority to launch (which depending on the rules of engagement may require going high up the chain of command), get fire control radar tracking the targets (something like SBIR wouldn't be enough, the missile battery needs its own high-quality track), and then launch the interceptor. Also add that people are human beings, and that no one was expecting an eminent strike on Qatar from anyone.

Qatar has a THAAD and Patriot installation, the former assumed to be manned by the US, which can engage these threats. The latter can detect them at a much lower range.

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.