While I agree with the great majority of this, two things…
If warnings simply appear in the inbox (and notifications), what would prevent users from simply never checking their inbox and later claiming ignorance? With the current warning system, users have to acknowledge that they understand the reasoning for the warning (whether they agree with it or not). This ensures that no one can say they “didn’t see the warning” later when appealing a ban.
Additionally, being able to browse the site during a ban, would defeat the purpose of a ban, which is to forcibly remove a user from the site. After a ban (assuming not permanent), you could then log in and check your assets to, as you stated, ensure that they are not also violating the rules.