Yes, they could but they don't have any obligation to do so since it is not "standard features" defined by any NIPs. Start with NIPs and convince (make issue, pull request, or bounties) relay developers (many implementation) to do that is the way. Since it is hard thus client developer(s) choose to monitor relay data on their own as solution by running custom software in server to support their client software.
AFAIK, Amethyst do that https://github.com/vitorpamplona/amethyst/blob/main/app/src/play/java/com/vitorpamplona/amethyst/service/notifications/PushNotificationUtils.kt . Other client like ZBD might do that. Not sure with other clients.