Signing the citation is a promise to appear. Then if the person cited fails to appear, the court can issue an arrest warrant for failure to appear.
These procedures are logical and straightforward and standard across jurisdictions. Do you really think you’ve discovered some shortcut here that all of the pros have overlooked?
No, those proceedures are not logical, because they are completely unnecessary.
Her signature would have changed nothing. A judge would in either case have the disgression of signing a warrent.
Police are losing contact with reality. Tasing the woman was aggravated assault, and possibly mayhem if the child were harmed. I’m sure they had far better things to do that this.