Does unlocking smart lock turn entire system off?

The SS door locks are designed to be integrated with the alarm system, so yes they turn it off.  There is no current way around that, but could possibly be updated later to add that.  Right now the easiest solution would be to buy another brand smart lock since you want them to be 2 separate systems anyways.

That's unfortunate and seems like a missed opportunity for a useful feature. If I bought another brands smart lock, wouldn't I still need a SS entry sensor on the door, and I'd still need to turn SS off whenever I opened the door? That seems like a rather convoluted work around.

Isn't that exactly what you asked for in your first post though?  You should have an entry sensor on the door anyways.  The smart lock will not sound the alarm if the door is kicked open.