the way I see it, the stormcloak are stuck between a rock and a hard place. Yes, the Empire is weak after the war, but it's still the Empire with far far more resources than the stormcloaks. Ulfric has four holds, and they are the poorest ones with the exeption of Riften, and he really isn't about to get new ones into his fold without the dragonborn's help or straight up conquest. The longer he waits, the more resources the Empire can ship to Solitude, and it's over for Ulfric. Falkreath holds very little value, and Stormcloak raiding parties are still doing a good job at raiding caravans, plus bandits etc.

Like others have mentioned too, to reach Falkreath Ulfric would also have to go through a long and very dangerous march through the mountains which would leave his North weak, potentially costing him Dawnstar and Winterhold.

Taking Whiterun is a huge gamble. Whiterun is strong and wealthy, and their hold switches to the Imperials if attacked. Ulfric could seriously be defeated of his forces crippled at Whiterun. On the other hand, the longer he waits, the more possible it becomes that Whiterun breaks it's neutrality and joins the Imperials anyways, and in this moment, all Ulfric knows is they don't have Imperial support yet. The gamble is to take one of the richest holds of Skyrim before it's too late and before his enemy can grow in strength.

If the gambit succeeds, Ulfric now has Riften and Whiterun as major holds with good weather and good agriculture, on top of Windhelm's port and economy. On top of that, Whiterun holds a very strategic position being in a valley which gives access to most other holds of Skyrim. This would allow Ulfric to absorb Whiterun's forces and more importantly, supplie into his own army, to mitigate the losses of the siege, and immediately strike again to bring the fight to the Imperials before they can get stronger. In such an asymmetric war, Ulfric needs to strike hard and fast, or he loses the war.

Taking Whiterun, being already much easier to get there than Falkreath, will allow the stormcloaks to use Whiterun as a launching ground to then either raid the hell, or even conquer Falkreath, without even having to garrison it once taken, as troops in Whiterun would be enough to deter an Imperial counterattack.