For example, if you had better response ability you would receive more difficult tasks. And if you are responsible you are response able, or able to respond.
No, it did not.
Etymologically, the word "responsibility" comes from an obsolete French word "responsible", itself coming from a Latin word "responsabilis", the past participle of "respondere", meaning "to respond". The word did not come to imply any measure of accountability until the middle 1600s.
The suffix "-ability", on the other hand, comes from the Latin use of "-abilitas" onto an adjective to form a noun.
The word ability, though, is surprisingly not derived from this use of the "-abilitas" suffix in Latin. It comes from the Old French "ableté", meaning literally "ability" or "capability".