I recall reading the same thing about the first battle of Bull Run in what I think was Jim McPherson's Battle Cry of Freedom (an excellent book, regardless of whether my citation is correct!) so, yes, it did happen. However, I have a hunch that this was a one-off. I imagine both Union and Confederate counterintelligence would look dimly on the idea of picnickers watching troop movements, and as the war took its toll on Southern transport infrastructure, a day trip to Chickamauga would have become quite difficult.

As for why this was attractive, there people weren't out of touch per se. I imagine very few of these Washington aristocrats expected years of carnage of civil war and did not eagerly anticipate to be camping out years later in the Wilderness, at Vicksburg, or at the Crater, even if that was possible. Neither did the Union Zouave regiments and the first wave of volunteers who, on both sides, often expected the war to be over quickly. In the opening of the Civil War, many on both sides boasted of a quick, easy victory. But even if it was physically possible for spectators to watch battles after Bull Run, I imagine few would have done once it became more apparent that the Civil War would become a long and bloody conflict.

Northern aristocrats did participate in the Civil War though, and not just through enlistment. I'd direct you to Matthew Gallman's Mastering Wartime and (even though it's a little dated) George Frederickson's Inner Civil War if you want to know more about how Northern urban elites undertook what they considered to be their patriotic duty on the home front through the Sanitary Commission, raising money for bounties, and buying war bonds. I suppose what I'm trying to say here is that the Civil War could only be a spectator sport when Americans thought of it as a short, easy conflict, and when that assumption was broken, aristocrats turned to new forms of participation more suited to the conflict.