3 minutes is a pipe dream, especially at smaller rinks.
She was exaggerating, of course, but it really is the preferred method to first notify the current coach before approaching other coaches about availability. On the other hand, if you start interviewing coaches first, they won't want to accept a student whose family doesn't show consideration for the current coaching situation. "You can be replaced" is the attitude that communicates and no one wants to feel disposable. At a smaller rink, you might never find a coach willing to accept you as a new student, as you know.
Is it uncomfortable to say "I think we're going to try another coach - just letting you know in advance."? Sure, it's uncomfortable, but it opens a dialogue and allows the opportunity to either salvage the current relationship, augment it, or to part company without burning bridges. All PSA coaches know what to do when a client says "I'd like to talk to you about this..." If a coach makes a fuss, barring unpaid bills, they have bigger issues than losing a student. If you're old enough to spend the money, be mature enough to treat people the way you'd want to be treated.
To address someone's else criticism: this has nothing to do with "ownership" of students. It's courtesy and professionalism.