Bilbo is, without a doubt, a terrible hero.
I know, I know, build him up with half a dozen attachments and he's good- but any hero is good if you do that. He literally has no good stats, which means all of his usefulness comes from the attachments. That means if you attached those cards to a hero that is already good at that thing, they're going to be better than Bilbo every time.
The usual advice is to build him up as a defender with cards like Protector of Lórien and A Burning Brand - but in the same sphere and for less threat (yep, actually less threat) I could have Denethor as a defender, who has +1 and . Bilbo will have to ditch a card to Protector every turn just to have the same defence as Denethor starts with, and he's more likely to die if you mess up. Suddenly the extra draw for Bilbo doesn't look so great.
If you want card draw and a hobbit, the obvious choice is Pippin at just 2/3 the threat cost. He'll most likely draw you more cards over the game anyway and you don't need to build him up so much, as the obvious place for him is questing.
The big issue with Bilbo though is opportunity cost. The time you really need your heroes to step up is early, before your board state is very advanced. Bilbo is hopeless then; he's not built up yet (it will probably take several turns to get him to "ok") so you are basically down one hero. He works against the hobbit theme of low-threat turtling to get set up, because his threat is so high. The passive +1 card is a trap - he'll use early card draw and resources just to get him to the level of other starting heroes, meaning that he is a net negative in both cards and resources.
I think Bilbo suffered from coming out so early in the game's development, and was just a complete miscalculation by the game's designers. If he came out now, I think he'd be only 6 threat at most (I personally think he should be 5). That may seem extreme, but his one stat higher than 1 is , and that is the one stat where it isn't very helpful to have a value of 2- the mechanics of defence basically mean that <3 often might as well be 0. His passive draw is not helpful when you need several cards to get to "break even" point, and you don't even get the draw yourself every turn.
Really, the only reason to use Bilbo is if you want to handicap yourself. I rate him as the worst hero in the game, and by some margin. The only question to me is, are you better off having only two heroes rather than taking Bilbo? And that is a pretty sad indictment.