Why are dogs routinely used in search and rescue missions?
Because search and rescue missions aren't usually performed with the aim of depriving someone of life, liberty, or property.
Why are dogs routinely used in search and rescue missions?
In search and rescue missions, false positives don't generally trigger violations of privacy rights.
In search and rescue missions, the dog's handlers don't want false positives; they don't decide that they want to search a pile of rubble and then bring in a dog to justify it. If there's nothing in the rubble, they don't want to waste time searching. By contrast, in many cases where dogs alerts are used to justify a search, the police desire to perform the search precedes the dog's alert. Even if there's nothing for the dog to detect, many officers would still rather have the dog alert so as to justify their suspicions, than have the dog remain silent and thus let the suspects go.