Live site is not by definition bad. You get to work on very real issues close to or directly with customers and need both technical strength to fix the problem as well as soft skills to communicate well across, up and out. What would be interesting to know is if the on-call loop is too small such that you are too frequently the active on call. Also, if the team does not have a thorough live site guide documented / if you need to beg a few secretive seniors to get the knowledge they should have written down long ago. Does the team write service improvements, tooling or does code refactoring to address live site issues trends?