Imma be honest, even with Kizaru holding back, with feats we can argue a fight against Kaidou and Big Mom would simply come down to Kaidou and Kizaru. Kizaru makes Big Mom a non factor in this, she can't keep up or tag him not only because he's the fastest in the verse, but also him being a true intangible logia the homies cannot stop him either, so she has no means of playing a significant part if Kizaru wants to play hit and run.
It's no wonder then that Kizaru was so confident given his abilities because remember Kaidou could not even perceive G4 Snakeman.
Taking down the 2 of them is different from handling business and separating them. He now has the speed feats to back his statement, especially when you factor in he never went all out.
No one is catching Kizaru if he wants to escape, so to answer your question, no Akainu did not save him he could save himself at any point by leaving if he felt threatened.