Accuracy
The obvious issue is that since speed of light is many orders of magnitude larger than the speed of sound, any inaccuracies in measuring the sound propagation time will have a disproportional effect on the calculated estimate of the speed of light, so a small inaccuracy in your estimate of speed of sound or any tiny delay in the process of determining when exactly "sound wave reaches the clocks" will make the whole endeavor useless.
Speed of sound depends on all kinds of factors such as temperature, humidity and pressure, so you would need very accurate knowledge of the atmospheric conditions (and very accurate information about their exact effect on speed of sound) throughout your experimental area - since speed of sound is approx million times smaller than the speed of light, a 1-in-million error in the estimate of speed of sound means +/- 100% error in your calculation of speed of light. The same also applies for your second scenario unless you can somehow control conditions ensure that the speed of sound is equal in both directions from the synchronizer, which won't be true by default.
Propagation of sound also is diffuse enough to limit you to relatively small distances (it's tricky to accurately transmit a pulse of sound over 10+ miles) so the delay of sound propagation across a part of the receiving apparatus can be larger than the elapsed time of light propagation you're trying to measure.
So the conclusion is that you technically could try to do it this way, but it's simply not the most practical way to construct an accurate apparatus to make this measurement useful. Like, as you say, having a "controlled air tight environment" removes some sources of inaccuracy, but it's not trivial to have controlled air tight environment that's, say, a mile long, and it's difficult to get an accurate measurement of the speed of light if you want to limit the experiment to something room-sized since the elapsed time would be so tiny.
And of course, if you would postulate that there's some difference in light speed one way over the other, then this would likely also affect the directionality of the speed of sound, in that case you can't simply assume that the speed of sound is equal in all directions. 795