Police need a warrant to get location data from your cell phone provider. No department is bothering with that for a welfare check unless the situation is really dire, and the process still takes a while to carry out. "Where do you think your brother is?" "Oh probably at XYZ's place" is a much more straightforward and common process.
I have some family who are dispatchers (not in the US), and they have the ability to immediately get location data directly from providers if they think somebody is in imminent danger. They can’t do it to track a suspect (not without a warrant), but they would regularly do it for suspected suicides, or when a caller communicated they were in danger but failed to give their location for what ever reason.
I wouldn’t be surprised if many jurisdictions around the world had similar systems.
In the cases I’m familiar with it was often used for other people calling in suspected suicide attempts (or other types of self harm, or drug overdoses, ect…)
I am a reporter who covers crime. Your account is 100% accurate. Federal law allows for the disclosure of customer records “to a governmental entity, if the provider, in good faith, believes that an emergency involving danger of death or serious physical injury to any person requires disclosure without delay of information relating to the emergency.” 5 U.S.C § 2703(c).
Either corporations are allowed to be 'pressured' by the government or they aren't. But of course the legal system will make themselves a nice carveout for their uses like always.
They may have done both. Or they may be lying. In most cases it's best to assume that the police won't give you any info unless they are compelled to do so (and sometimes not even then).