One think I just realized is- why is Link even chasing Abigail? One idea is perhaps Link knows from the letter he stole that Abigail is going to kill the Shiekah elder, the wisest of their leaders. Link, feeling guilty that he got the Gerudo the arrows to get to this point, goes to stop her. When he arrives on the scene, she is just about to depart from her troops to find him herself, and Link chases her.
Then, for victory, this is what I was thinking:
As you're chasing Abigail, for example, she'd run across a bridge, which might collapse, so you have two paths to continue after her, you could knock out a Gerudo who's firing arrows at some Shiekah and cross the bridge she was guarding, or shoot a fire arrow at a shiekah blockade to burn it out of your way. At the end of the chase sequence, the game would tally up which side you've assisted more.
Then, the end of the chase would be altered either way. You catch up to Abigail as she finds the Shiekah elder? If you helped more Gerudo, then the elder will call upon his body guards, and Abigail will say, ha, it seems they've busy losing the battle. Or in the other case, the elder will say, oh so you came alone? Or did all your reinforcements lose to my warriors? Either way, they'll start to fight, and Abigail will shoot a fire arrow Link's way, burning down the path to get to them to interfere.
Link will have only one other path to take, and down it lies the entrance to the temple, and Seere guarding it. He says his line about "you could bring salvation or Hyrule- or damnation, and pushes Link into the temple and seals the entrance. More on how the entering will work in a minute.
Then when Link beats the boss and comes back out of the temple, Seere is standing at the temple's entrance, looking down on the battlefield. Whoever's side you helped more, that leader (Abigail or the elder) has just defeated the other in battle. Their second-in-command is talking to them. Seere explains the situation to Link:
[Whichever] has been defeated, but both sides suffered so much loss that the Shiekah can't hold the desert pass, but the Gerudo can't advance anymore and are retreating. The Shiekah city/fortress is in ruins. Seere says something like "I don't blame you for this, you just added a spark into years and years of kindling. In fact, I'm still not sure whether you're destroying Hyrule or bringing about the healing we've been needing for ages. But that's just me- these guys, they definitely think you're to blame." And several Shiekah warriors jump into the scene. Link has to escape the ruined Shiekah fortress to lose them.
So that's my opinion that the choices you make should only alter the plot slightly. Anyway, about the temple's entrances, I think we need even more background.
The Shiekah's job is to keep the Gerudo out. I imagine they do this by occupying the only pass in/out of the desert. Perhaps there are secret ways only Link can take, but the Gerudo people can't leave unless they go through this Shiekah fort. I watched Full Metal Alchemist Brotherhood recently, and if you're familiar, it's a concept like Briggs fortress in my imagination.
Link isn't even thinking about going past this fortress into the desert when he goes to the Sheikah fortress to find the cog, but due to interference from the Hero's Ghost, link finds his way into the desert. As he comes back with the Gerudo's attack force with assistance from the fire and ice arrows, he comes into the fortress, which also guards the entrance to the Shiekah temple.
So, he's sealed into the Shiekah temple, along with the monster that's been creeping around in there forever. The monster attacks Link shortly after Link enters the dungeon, and in trying to escape, Link finds a secret exit out of the dungeon, which leads to the Hyrule side of the fortress. The only way back into the fortress is through this secret passage, and you can't even get back into the fortress, because the main entrance to the dungeon is sealed.
Does that make sense?