"If an army is too large for its transport network, what happens?"
I would think that the central Tennessee front - the Army of the Cumberland - would be the most obvious and distinctive example of a theatre in which Northern armies were severely limited by logistical considerations. The armies along the Mississippi, the Army of the Tennessee, and the Army of the Potomac all could rely on uninterruptable riverine lines of logistical supply. Meanwhile, south of the Cumberland line, the Army of the Cumberland had to rely on crazy-inefficient wagon trains, which greatly limited that army's strategic mobility.
The worst operational collapse of the Army of the Cumberland was the campaign that culminated in the Chickamauga debacle. They were caught dispersed, and barely consolidated before the collision of forces. But they *did* consolidate, and the defeat was due to poor tactical command-and-control - Rosecrans acting as all three of his corps commanders instead of trusting them to fight their own corps - more than operational dispersion. But the Army of the Cumberland wasn't "too large for their networks", it was too large to be fought as a single corps.
The problem with positing small, professional divisional-sized armies which can strike free of the big logistical corridors, is that they can't accomplish anything permanent when they get to their targets, which tend to have organic logistical connections with the rest of the network. They're glorified raiding forces, which can only do the flashy but transient damage of a raiding force. Look at Jackson's and Early's plunges into the eastern theater's strategic rear. Glorious, flashy slashing about, but they generally ran out of room to run and either withdrew, having done nothing lasting, or got bloodied by big, dumb forces moving into blocking positions via the rail and water lines. Even at Second Manassas, Jackson would have been destroyed by Pope's flailing about if he hadn't had Lee's mass army racing to consolidate beside and behind him, and make it a victory.
Lastly, consider the first Corinth offensive, with its lack of fighting, and slow, lumbering advance "in touch" by three massed forces against a smaller defending force. This describes your earlier "pools of forces" clustered about logistical LOC. But the end of the campaign, taking the Jominian strategic point, left the attacking force with nothing further to do, except to disperse in search of new goals, and to get the troops away from the medical disaster which was brewing there.
In short, you can't rely solely on small, strategically mobile "professional" armies when the opponent has massed armies and can rely on quick lines of communication. Any sort of attritional contact uses up your professionals, while the "massed" enemy can replace his losses. Early, for instance, with his crack, veteran, near-professionalized Valley army, still was defeated again and again by Sheridan's masses, in a logistically isolated theater like the Valley.