That would be an excellent way, but usually it is just them being excellent buddies for no reason. It is an error so many fanfics run into, that they are too glued to the main cast and want to be friends with everyone. Which is fine, but don't try to sell me your self insert as a bad arse whatever then.
...but isn't the purpose of the self-insert fan-fic to murder the (former) protagonist? (and steal his cat-girl, of course, but that subplot isn't allowed on the ... certain sites)
That's because of a copyright law though. If you pay the appropriate sum of money to the IP owner, you could violate the canon as much as you like. Case and point: Amazon's Ring of Power.