No, a free-ballooning (sail powered) non-magical airship won't work.
For two reasons, and you may be surprised by the reasons.
First, yes the airship can steer...a bit...by varying it's altitude instead of by using 'sails'. Using favorable winds at different altitudes is how free-balloonists have steered for over a century. Sails don't work for steering - there is no other medium for a keel or rudder to push against.
However, there's a big difference between balloon-style steering and actually reaching a destination. Powered dirigibles often needed to make multiple approaches to the landing field due to an unexpected crosswind or navigational error, and a powered fly-around could easily take 30 minutes. An unpowered fly-around might take hours...or be impossible entirely.
Aside: Non-magical airships cannot just drop into a cow pasture in an emergency, either. That's called a 'crash'. Ground crews of 40-50 were needed to walk larger Zeppelins into the mooring mast. Airships may be buoyant, but have a lot of inertia and sail area. Several ground crewmen were swept into the air --and killed by the fall-- by minor errant breezes. You can use heavy tractors today instead of large crews (Goodyear did), but you still need to get the airship and the tractor close enough, and it's still dangerous. Remember that this means the non-magical airship must arrive at destination at least 3-4 hours before sunset to be landed safely...unless your landing field has hectares and hectares of non-intrusive illumination and a ground crew trained for night landings.
The other reason a sail-powered non-magical airship won't work is that they are simply too big and fragile to survive ordinary weather for long. Most non-combat losses of airships were due to simple weather. Example: The USS Shenandoah was literally ripped apart in midair by an otherwise unremarkable summer thunderstorm over Ohio. Powered airships generally cruised outside of a weather front, searching for a weak spot to punch through, The Graf Zeppelin once spent 6 hours over the South Atlantic scouting (successfully) for her safest course of action through a rather mild front. A free-balloon simply lacks that ability, is likelier to be sucked into the worst storm cell nearby and promptly destroyed.
One other major drawback of unpowered non-magical airships, though not a critical fault, is their inability to maintain accurate navigation. Powered airships, before LORAN and GPS, relied upon celestial navigation and detailed ground charts. This meant popping above the cloud deck several times each day while over the ocean to get a sun/star position fix. Powered, this is a quick, accurate operation. Unpowered adds time and positional uncertainty to the journey (how far off were we blown by those five layers of crosswinds?), increasing the likelihood that the airship will miss it's destination and need to (sigh) fly-around.