[adgjlsfhk1]

The trick is that if none halt on `b` steps, you don't know that BB(x)<b. Specifically, if you have one TM that keeps going, you don't know whether that TM halts eventually or keeps going forever.

[josephcsible]

Sure, you don't know whether BB(x) < b or BB(x) > b for that reason, but wouldn't you still know BB(x) ≠ b, and isn't that good enough?