You can't get what you want...
First off, the reason why your latchset's behavior seems quirky to you is because it's a storeroom function latchset, where the key is used to open the door from the outside, instead of relying on the handle to open the door. Yours likely has an anti-vandal (Vandalgard™) function on it, as well, which would explain why the outside handle moves but does nothing.
That said, the behavior you want isn't something you can legally have. You see, the entrance doors to multifamily dwelling units (apartments and condos alike) have a rather important job to do, namely holding back smoke and flames in case the building catches alight somewhere. However, they can't do that if the air pressures and breezes generated by the fire simply push them open, so IFC 705.2.4 (and NFPA 80, the fire door standard) require that the door latch closed whenever it shuts:
705.2.4 Door operation
Swinging fire doors shall close from the full-open position and latch automatically.
This means that your plan to "dog" or "hold back" the latchset on your door and rely on the deadbolt as the sole locking and latching means is a dead end, smart lock or no.
but you can get what you need
However, this does not mean you cannot have a smartlock on your door. While Schlage does make the LE series of products that provide a one-for-one replacement for your existing L-series mortice latchset, given that you've already gone a different route, there is still an option open to you.
In particular, you need to convert your existing latchset to a passage set that doesn't require a key to open from either side. This provides you with latching functionality that's independent from the deadbolt while leaving the deadbolt as the sole locking means. This then lets you fit the August to your existing deadbolt. It'll require what's basically a swap of the guts and the exterior trim on the latchset, though, so you'll need to have a locksmith do it for you.