Yes, the Host Status alert should cause the device to be marked as "Dead" which should suppress all other alerts on that device. However, there's often a race condition if other thresholds are set. The Host Status alert threshold is >300 on the idleInterval datapoint. It's definition is:
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The interval in seconds we do not get data from the host. NOTE: there is server side logic that declares a host DOWN after 6 minutes, suppressing other alerts. We do not recommend you change this alert.
So, the alert opens at 5 minutes, then the logic kicks in to mark the device as down at 6 minutes. If any alerts open before that, they are not closed, and there was nothing stopping them from opening.
I know there have been a few requests allowing customers to modify the built-in server-side logic that marks a device as dead. I recommend you reach out to your CSM to add your support to that request.