Here's solution that requires the Terminal, but it works with Big Sur. This is useful in particular if you have dozens of notifications from the Finder that can't be quickly closed using other approaches.
From Terminal:
ps -e | grep /NotificationCenter | grep app | awk '{print $1}' | xargs kill -9
This is similar to Joey's answer, except it does it all in one command. ps -e gets a list of running processes. The grep filters the list to those containing /NotificationCenter. This list still has too many items because the grep command itself ends up finding itself. We exclude it by running another grep on the output of that list. The awk command prints the first column, which is the process id that we need to restart. Xargs takes the output of the previous command and feeds it to its arguments, namely, "kill -9" which is the unix command to exit a process.
The practical outcome is that the NotificationCenter.app is stopped. Launchd (the system process that launches other system processes) notices that Notification Center went away, and reinitiates it. When it comes back, all the messages in its queue are lost and your notifications are cleared.