Are owners discouraged from making multiple sitter requests at the same time?

Yes, when a person uses Rover to look for a sitter, the system immediately encourages them to send more messages continuously to increase the chances of finding the right match. Many sitters respond timely, like yourself. Yes, it will affect availability if not booked or archived (a good habit may be archiving within 24 hours if not booked, because you can always remove from archives and book later).