I fear some of us confuse the term "counter steering" with something else.
Yes, you must lean in order to turn. No way around it, unless you're at crawling speed, point at which, you can just turn the handle bars without leaning (you are still leaning, but since you're going slow, you can deem the position as vertical).
Countersteering is when you push forward, not downward, on the side of the handlebar of the direction you wish to turn. Pushing downward causes the bike to simply lean in that direction, which is fine, but that's not countersteering. Pushing forward on the handle bar causes your front wheel to turn in the opposite direction of which you wish to turn, causing the bike to fall into a very quick lean in the direction you wish to turn. There is no way for anybody to lean the bike as quick to one side or the other as quick as countersteering.
Countersteering is not the only way to get the bike to lean, but it is the fastest. You cannot push the bike down in a split of a second, but you can by using countersteering. The need for you to lean the bike quickly rises exponentially with speed. Why would you have to countersteer at 5mph? You have all the time in the world to lean the bike when you go slow - when going fast, countersteering becomes essential.
By the way, if I misread and/or misunderstood your post, I apologize. I'm really nothing more than a rookie, and I don't mean to be lecturing anybody. I just wanted to clarify the topic, to be sure we're all talking about the same subject.