According to the on-hit effects page, lifesteal is applied to all on-hit physical damage. Is this wrong or is BotRK's first passive just useless? ("Life steal applies to damage dealt by this item's passive.")
That line is very likely a relic, and I think it can be removed safely. There used to be a time where BotRK's on-hit damage specifically didn't trigger life steal, and that was then changed, so it's likely its pre-change state was just a special case.