In the few games that I've played, I'd say yeah. You've got a ton of orcs and southrons that are ready to assault the place, and Gondor in general has a lot of victory points. You'll probably want to get the Witch-king out as well, so you're looking to spend a muster die to get Sauron to war anyway.
The flip side is that if you don't attack, it seems you risk really bad stuff happening. A common strategy is to spend an early muster die to bring Gondor one step away from war on the political track, so when the Shadow attacks Osgiliath, Gondor is pushed to war and the FP player has a brief opportunity to muster an elite in Minis Tirith. Another common strategy for the FP is to get Gandalf the White out in Fangorn pretty early for the extra die. If you don't put Minas Tirith under siege, there is a real risk the FP will move Gandalf the White to Minas Tirith (it's 3 spaces away from Fangorn) to activate them (if they haven't already been activated by the witch-king), spend a muster die to bring Gondor to war, and then start mustering troops. At that point, you may be faced with a real possibility of a FP army on the offensive, with Gandalf the White leading (and Dol Guldur ain't far). Secondly, by leaving Gondor unscathed, you allow the FP player to easily bring Aragorn into play, which gives them another action die, and grants them another strong leader with which to attack shadow strongholds.
Really, it's probably best to attack them. Many players (from what I can tell) advocate attacking them first.
Edit: I just read your big defense problems. You can typically avoid Gandalf the White being in Minas Tirith by not bringing in a minion until it is under siege (indeed, I rolled a friend who brought out the Witch-king early, and I snuck Gandalf the White in before he could get Minas Tirith under siege, and with him alone, it wasn't pretty). The free peoples must have been motoring the fellowship for Aragorn to get there. If they start throwing people in there, the best strategy is to put it under siege, and then leave it. That will prevent those characters from creating difficulties elsewhere.