I've passed your feedback along to the developers! If something like this is implemented, it will likely be in V3.
Deleting Chess.com Messages
Yeah, can we please have an option of not being spammed with messages about having a paid subscription? It's incredibly annoying.

It's an old post but if anyone feels they want to delete more messages maybe make a suggestion to staff.
https://support.chess.com/article/295-how-do-i-report-a-bug-or-make-a-suggestion
Hello, currently the site allows you to delete 15 messages at once. This seems just a bit too low. It should allow 100 or if not 500 of them at once.
i personally have 7109 messages due to team match allerts from ChessCom and I'd like them to not crowd that inbox.
Is it possible to increase this amount?