Considering the logistical disparities between the two sides, I don't
understand how either of these strategies could have worked.

Kinda hitting the nail on the head there. There really wasn't a strategy that would have been effective without substantial outside help (i.e. allies), massive failure on the part of the Union, and/or extremely effective leadership and logistics from the Confederacy. Even on paper, the Union was gonna win, but that was assuming the Confederacy would actually be able to operate effectively as both a nation and maintaining a fighting force across 2000 miles of front. Supply quantity issues, "States Rights" problems affecting what soldiers got what stuff, lack or manufacturing/wrong places of manufacturing, ineffective leadership, internal factions and infighting, etc. all caused internal problems that by themselves would be sufficient to doom the Confederacy.

But let's say for sake of argument that none of that was a problem. The CSA can produce enough food, tents, blankets, shoes, weapons, horses, uniforms, medical supplies, etc. enough to keep each fighting man equipped and fed decently. The idea of forcing the Union to conquer space seems like it could be effective, a la American Revolution style. Make the enemy spend resources and men taking territory, and bleed them. The big difference from 1775 to 1861 is that the Union is not 3000 miles across the ocean. Their supply lines are just up the train track a few days. They have telegraphs to communicate in minutes. And they have vast resources to put into the fight, with a strong patriotic push from the nation. Tens of thousands of men volunteered to be a part of the Union war effort. It may have taken longer, but the Union would have conquered the land of the Confederacy either way. As Forts Henry and Donelson, Corinth, New Orleans, Columbus-Belmont, Vicksburg, and many more showed, the Confederates were incapable of successfully stopping the Union from taking their territory and transportation hubs. Plus, if the Union continues to seize territory, what is the incentive to give up? You are clearly winning. So that strategy, while slow, would not have worked.

Lee's strategy was more of a "big gamble-big return" type. In the short-term, this was likely the only possibility, however slight, for a Confederate win. But ONLY in the short term. Had Lee been able to take the fight into the north in September 1862, destroy another Union Army on his own chosen ground, and begin to wreak havoc in the North toward Washington, maybe, and I emphasize the minuscule possibility here, he could have convinced Congress to sue for a peaceful cessation of hostilities. But again, only in an extremely short time frame. Any longer than a few months, and he has an army in Maryland or Pennsylvania in winter time with no solid lines of supply. If it could have been effective, the time frame for everything to go as near perfectly as possible was razor thin.

In any case, Lee's strategy, as well as his tactics, though effective in terms of winning early battles, led to devastating losses that could not be sustained. He gambled big every time. And while sometimes betting it all on red will net you the win, eventually that strategy will bleed you dry.

So to your last question, we have to understand that at the beginning of the war, everything is solely on paper. On paper, the Confederates have a President who is a West-Point educated, combat veteran, with loads of political experience as well as having been the Secretary of War. Against the single-term Representative with no military experience. While the Union certainly has the numbers and resources, its not known whether they will be able to get men to fight, or be able to actually use their resources effectively. For the Confederates, they believe firmly, as so many of them stated repeatedly, that numbers wouldn't matter because Southerners were better fighters. They also recognize, just as the Federal generals do, that there are all kinds of ways to mitigate number disparities. In 1861 there is no way to 100% know what is going to happen. You rely on what you believe will be your advantages, and hope to minimize or mitigate your defaults. We, in the 21st century, can know what Lincoln thought, what Congress was going to say, how the nation would react, the Emancipation Proclamation, as well as how ineffective a leader Davis would be, and how poorly ran the Confederate supply chain would be. But in 1861, neither side can know that about their enemy.

In other words, every nation goes into every war based on assumptions. You assume you can win, so you put forward efforts to try to achieve that. You also assume its possible to lose, and take steps to avoid that.