I noticed that the notifications were only sent if a user is visiting Discourse every day, and will not get any emails. Would it be possible to send out emails for every notification the user is receiving?
No, mailing list mode means that you receive email from everything, except categories you manually put in “Muted” mode.
If you want emails only from watched content instead of mailing list mode, you can enable it the Emails user preference setting named Email me when I am quoted, replied to, my @username is mentioned, or when there is new activity in my watched categories, tags or topics by setting it to always.