To begin with, I do not think that it is a question of character that "qualifies" or "disqualifies" a Ringbearer. Bilbo simply picked up the Ring and slipped it into his pocket. I don't suppose Prof T is justifying the "Finders, Keepers" maxim, but Bilbo never did have an opportunity to return the Ring to Gollum, (who did bloody murder for it, by the way) so I guess you can say that he is pretty much entitled to keep or "bear" the Ring. Subsequent transfer of the Ring from Bilbo to Frodo and then to Sam is perfectly justified.
If there is one quality that "qualifies" a Ring Bearer, I suppose it is the legal transaction by which the Ring is obtained. Taking one situation for example, during the Council of Elrond, Master Elrond placed the responsibility of destroying the Ring to Frodo. Is there really any other characters that can possibly contest that decision? Yes, Bilbo has borne the Ring before, and he had all the "Elvish cum Hobbit" quality aforementioned, he is also a more experienced adventurer. But Gandalf told him that the Ring has "passed on". With Sam, it is even more interesting, since Sam Gamgee loved Elves, but he would prefer his tatters and tomatoes to travelling with an Elf any day. Then again, one could hardly observe anything Elvish about Sam until his return to the Shire.
Besides, Frodo failed to destroy the Ring in the end, didn't he?
P.S. Yessss! My Finals are OVER! Back to La-La land!