Yeah, that's one of the parts that made me go ''qué?''. Softened!Leliana almost gets rid of all the Chantry's accumulated baggage and prejudice overnight. just by being nice apparently. I'm not sure if it came across as more saccharine than the writers intended, or if it really is the perfect, ideal solution where everyone miraculously holds hands and sings kumbaya together. I hope it's the former honestly, makes for a more interesting story than ''see, being nice solves everything kids!''.
She meets with their leaders, urging unity. Miraculously her words take root, and - for now - the Chantry remains strong.
This line of dialogue is incredibly vague and I can see where you'd think that Leliana just had an afternoon talk with the rival sects and boom, peace, love and understanding for everyone. The point is though, we have no idea how long these talks were. They could have taken months or even a year of constant negotiating, debate and peacemaking, which I assume to be what happened because there's no way that Bioware wanted us to think as you suggest, that she managed to solve the Chantry's baggage immediately after she was made Divine. So, no, I doubt that were meetings with these sects were effortless or were expected to be so.
You also have to factor in that the Chantry at this point really has nothing to throw at Leliana if they do decide to rebel. The Templars either no longer exist or are no longer under Chantry control, the Mages obviously won't support them and are likely to do whatever they can to keep Leliana in power, and the Inquisition, of course, has Leliana's support as well. At best, they can get a few small revolts started from the peasantry or conservative nobles, but that's about it.