In the case of the MO/IT stockades, you start right next to them, with mobile units and active leaders, it is summer and you typically have decent weather, so the CSA can easily beat the Union to them and face auto-garrisons at worst.
It isn't like you have a choice to attack depots in IL or IN with these guys instead: Fayetteville is hell-and-gone from Illinois Farm Fields. So if you decided to leave the stockades in place, what else is there to do with a few cav, Indian and mounted volunteer units that is better than gaining medium-to-long-term security for your supply line?
Athena has never rebuilt the stockades in any of my games. A human can choose to rebuild, but stockade cards are a limited resource, so that's still at least an even trade, IMO. As the Union I contest those stockades as best I can and am willing to spend to rebuild in order to maintain access to Fayetteville from that flank.
As far as attrition goes, the reason to destroy those specific stockades is precisely to make the attrition and supply system work in my favor: any sneaky Union attempts to capture or destroy MY depots at Fayetteville and Springfield become substantially more difficult since it is THEY who now have to march without supply or shelter in order to get near enough to threaten me. I am trading a (very) slight cost in time in order to increase the long term security of a critical and very tenuous supply line. Within three or four turns from initial activation, they can accomplish this task and be on their way to Springfield to help in repelling Lyons' inevitable attack or begin the counteroffensive on Jeff City. (Tip for new players, ALWAYS blow the depot at Rolla as soon as the militia there activates or else the Union captures it on the next turn and Lyons stays in supply all the way to Springfield.)
Now, I grant you that the Missouri Game is a long-term resource sink if you can't convert and actually capture St. Louis. BUT, you are saddled with a lot of starting force in AR that is otherwise useless due to relocation costs, so you might as well play for St. Louis, which is a very tasty prize. Assuming you can fend off the early attack on Springfield, (which the stockade tactic directly helps you to do!) you will ALMOST have enough troops available in-theater to take St. Louis by main force, so it is important to grab every advantage you can.
St. Louis is not decisive of course, (DC is the only decisive objective) but it is a very big win for the CSA, and you can make it happen relatively early if you play with commitment.
I would posit that in terms of cost/benefit, St. Louis is the CSA's most achievable/valuable target after DC. It is also fairly easy to capture Louisville-Lexington-Cincinnati, but I think the CSA stands better with St. Louis if they can't do both (while not losing in the East).
Every other theater is irrelevant if one side or the other can win in the East, and All East is (IMO) the best path to victory. But if neither side can gain a decisive advantage in Virginia, then the side that goes out and kills the most enemy divisions in the other theaters wins.