But, the second the silver leaves Markarth, the Empire is free to pillage it. That's the thing with being surrounded.
Exactly. The walls of Markarth may make it hard to invade, but it's not like he can easily move the silver out of it either. In addition to the Imperial controlled Falkreath and Solitude, and the technically Imperial (but staying out of the conflict) Whiterun being the only areas accessible by the Reach, he also inherits the problem of the Forsworn, which now raid his supplies instead of the Imperials.
Simultaneously, the Imperials gain Riften, which is on the border of Cyrodiil, making it easier for them to move in supplies and reinforcements. Not only that, but they then have up to three (if Winterhold is given to them) provinces bordering Eastmarch, allowing them to attack from all sides. They also gain Maven Black-Briar's resources, who we know is more than willing to do things of a questionable morality to secure her station in power, meaning she could be persuaded to have the TG specifically target the Stormcloak caravans and whatnot.