Good luck leaning a bike in motion without turning the front wheel.
Having said that, it's 100% the case that a 2 wheeled bike with rounded tires in a lean will turn. Just consider that the circumference of the tire shortens as you approach the edge, it must travel a shorter distance per revolution. so without sliding, it clearly needs to turn.
In-fact, at speed, the lean of a bike is the primary factor that contributes to the turn.
Now, for wider turns, you do need to turn your front, but even then, it's usually in the opposite direction!
Anyway, on snow mobiles, you really don't want to need your steering handles, as they don't control a wheel, but skis! And staying on the trail mostly involves reducing speed and, well, leaning. By the time you really need your steering handles, or your brakes, you're really already in trouble...
In all likelihood you are doing it unconsciously, generally, as one goes to initiate, say a right-hand turn, as you begin to lean right, you also apply a slight additional pressure to the right handle bar, which steers the wheel to the left, this initiates counter-steer which leans the bike to the right. With a stable bike, and a fixed front fork, it would be exceedingly difficult to initiate a lean significant enough to navigate most roads at the speed that we do.
If I forget to unlock my front wheel, I can go about my ways just fine: it's just unpleasant.
I also ride snowmobiles on a regular basis, and if you require any effort at steering (like at braking!), that's pretty much an error: you shouldn't need to do it! And that's even at 60km/h...
Also, I'll be honest I haven't ridden a snowmobile in so long that I don't really have a mental model of it. I'm interested though, Do you think they are a similar experience? Don't they normally have like many more points of contact?
Still, the experience is exactly the same as a 'manual' bike (here's mine: https://www.gazelle.nl/puurnl).
Point is: you need to really steer beyond leaning? You either hit a over-40-degree turn (not uncommon even in Amsterdam!) or you did something wrong. You need to brake? Either an idiot crossed your path, or you definitely did something wrong!