Houthis use long range drones and cruise missiles which can easily be fired from the West over the Red Sea. There is no plausible situation where they simply don't point the radars that way.

There is also SBIRS and radars across Saudi Arabia. By far the likeliest scenario is that the US detected the attack and made the choice not to defend Qatar.

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.

Your second sentence is part of the answer to your first one...

You'd need air launched ballistic missiles (which Israel used) to avoid detection by Saudi radars. The Houthis don't have those.

You'd need jets capable of operations over the Red Sea - which the Houthis don't have.

If they did have those things, they'd certainly use them to attack Israel or Riyadh, not Qatar.

Except that Qatar's own systems also did not issue any early warnings.