I recommend 多功能成语词典 which is available as a (not free) add on to Pleco. It gives this explanation for 全力以赴:
This makes it clear that 赴 in 全力以赴 carries the idea of movement / signifies "to go".
(I think L Parker's answer is the best so far, but hopefully this comment still is worth putting up as an answer as it has a ...
Looking for logic in language? Likely an long look!
Why is it called 赴 fù when it is composed of zǒu and bo or bǔ?
zdic.net has this for 赴：
2 投入（某种境地），参加（某种行列）：赴战。赴敌（加入对敌作战）。赴难（nàn ）。赴义。
全力以赴：throw oneself into the fray with all ones might
This involves inversion (倒裝) in classical Chinese. The most sensible arrangement in modern Chinese is
To go towards / approach (sth.) with all one's might,
which obviously suggests the meaning 'trying one's best in doing something'. The grammatical object is the target but is omitted (perhaps understood) in the idiom.
While 赴 is transitive, 全力以赴 ...