My DD sounds an awful lot like OPs DD and like maestramommy just posted, she learned to ride on a comically small bike. Specifically, she learned on a 12" trek mystic at age 5.5. She rode that the first summer and then we moved to a 20" bike the following bike season. The small and light bike was key to getting her comfortable with learning to ride. She had a balance bike from age 2 on but never really took to it and I think she ultimately learned balance from a razor scooter. All they really need to know about balance to learn to ride is that you need to be moving to balance. Learning on a very slight downhill helps keep them moving. DH held on to her shirt lightly while she was learning but once she was willing to ride she was basically able to. She rode three biking seasons before she ever fell so I don't think falling is a foregone thing for a truly cautious child.
DS is a very different child and rides the balance bike well. He learned to ride a 16" bike at age newly four with the same holding shirt and running alongside. He can bail off a crashing bike without going down with it and definitely takes a more reckless approach to biking, although even with him the confidence to ride is a big piece of the puzzle.
momma to DD 12/08 & DS 3/13