This is toxic masculinity, and the idea that men have to be successful, strong, brave, etc. hurts everyone.
Being 'soft' or 'weak' should be perfectly normal and accepted human traits. How about we allow men the space and patience to access and acknowledge their emotions, instead of placing some patriarchal ideal of how they have to be big, strong, rich, successful, dominant etc. on them?
Easy: leave all emails that might be important as unread. That way, you know they're important because they're unread. Continue to never read them until you've got over 400 unread emails in your inbox. Then just mass delete any more than a month old, because they can't still be important or relevant. Continue to feel guilty and not read emails.
Absolutely flawless system for me so far.