OH please. Its a reasonable ask, and Im not 100% sure that exigent circumstances doesnt get them a pass anyway. Look at iit like this YOUR mother is missing, likely dead and the PD has photographic evidence of a suspect at her house the night she went missing. The subject is masked but it still shows some recognizable features. Hes armed, so the PD wants to know if he looks familiar to you. They have some folks who matched the video, and asked if any of them made a recent purchase. It doesnt seem too much of an ask.
Exactly what are they asking that hurts any of us? If they find some possibles, they interview them. Some will get weeded out others will remain possibles. Whats the problem? Especially when its YOUR mother.
They arent asking for a total record check, just recent purchases from a few possibles that fit the description. I can pretty much assure you that IF they had been forced to come back with a warrant, the search would have been much deeper