You are running your own business. All Rover does is provide a platform for us to conduct business, as well as providing some tools to use in our business. They do not follow up or monitor requests, except when certain key words (like cash) are used.
I give the requesters 24 hours to respond to my initial reply.
There are a lot of reasons why potential clients don't respond after their initial request. As Walt said, they are prompted to contact several sitters and they may have chosen another one. They may see your prices and not like them, they may have found family/friend to keep the dog.
When i don't receive a response to either my initial message or after a M&G in 24 hours, i send a reminder message in the morning. "Hello (client), just wanted to check and see if you still want me to care for / walk Fido. My calendar is starting to fill up and I want to give you first priority. Please let me know by end of day today what your plans are. I cannot guarantee a spot without a M&G (or payment)." If not response in 24 hours, I archive as 'owner didn't respond' and move on. By archiving, the request is removed from your calendar.