Where do you see it telling you you need precise location to see emergency alerts?
Your phone has two sets of things that could be called “emergency alerts”. One is the emergency alert system that’s controlled by the government and managed by your phone company. That one doesn’t require precise location.
The other is “crisis alerts” which is Google basically running a search for crisis near you and then telling you. This one may require more precise location.
It’s entirely possible for your phone to just not get the cell network based alert. You can be connected to a tower outside of the alert area while someone right next to you is connected to one inside. Or you can just not get it because cell communications are imperfect. The issuer will typically resend several times to try to ensure it gets through to people, but it’s not perfect.