Didn't literally say anything about killing him. The guardian from an island (I think its called Scotch but don't ask me where did I get the name from) told Drake something like "are you sure you want to mess with this? This is one of Kaidou's favourite islands" to which Drake answered "That will makes things much faster". So it could jave been a way of geting Kaido's interest by defeating an underling...? Or yeah, maybe he thought about attacking him but he tasted the same medicine Kidd and the rest tasted... and. changed his mind (although this doesn't fit too much with a character as Drake seems to be)
In my head canon the second option seems far more likely. I mean drake quit the marines to see what life is like on the other side. So he goes and joins kiadou? Ok saying hay out loud actually makes it make a little sense.
Sorry but I only ever cared about drake for 10 days. 7 when I saw his ancient zoan, it took me till the next chapter to remember ancient zoan is still zoan and therefore still fodder. (Like being king garbage is still garbage) and 3 days after he defeated caribou, I was stoked until I remembered that still doesn't make him too strong.