Maybe "self-insert" is the wrong word, but he's clearly designed to serve a strictly secondary character function like many love interests, but with the added burden of being reintroduced and newly promoted exclusively for that role - something good long term love interests lost a long time ago, and that often makes "throwback formulas" like the one Bernard is experiencing (outside of the bisexual aspect) more of a liability than an asset. More modern romance formulas require *both* characters to be engaging and for the relationship to make sense and work form both sides, meaning even initially secondary romance characters can become stars in their own right. Compare with how Rey in Star Wars was
reduced to being Kylo's fangirl, and contrast with how Stephanie quickly
grew beyond that romance role.
Fanfiction writers often focus on simple, older formulas rather than the more complex ones that we expect from professional writers - thus, fanfiction and fanfiction-style romances tend to be more myopic and bland on average (Tim and Bernard), and can become more toxic and disgusting at worst (Reylo).
Right now, Bernard is largely a strictly secondary character for a bland romance, likely because the writers themselves aren't really used to a bisexual romance and ar e playing it painfully safe.
Also, when it comes to TimKon, that's just the old cliche of a bromance being easily reconstrued as romantic in part because Western culture tends to be uncomfortable about close male relationships period. On the one hand, it doesn't mean there's no reason for the 'ship - there is an actual deep relationship already present, after all - but it also does play into how hard it can be for some audience members to see platonically close male relationships versus platonically close female relationships.
(yes, I know "LesYay" is a thing, I'm just noting the different cultural perspectives of it.)