I think it depends on how you view the issue. If you think about it, it was fate, because what was the
real reason Shri Ram was born? To be king or...something else? Only 1/4 of it was because King Dasharath performed the Putrakameshti Yagna. Darsharathji desired a son, but why did Lord Vishnu himself have to come as his son? Anyone could have been born as Dasharath and Kaushalya's son, right? Any of the Devas, Gandharvars, Kinnaras, etc could have been born as Dasharathji's son, but why did Lord Vishnu himself come?
That's because he, and only he could kill Ravan. There were many great and strong people and sages who had the strengrth to kill Ravan, but because of curses, it fell on Lod Vishnu to kill him. The curse of King Anawrata of the Ikshvaku Dynasty said only a person born in his family could kill Ravan, and you all know there were lots more curses Ravan had, so bascially, what I'm saying is, only one born in Dasharathji's house could kill Ravan, and also, Lord Vishnu, so you see, Lord Vishnu had to take birth in the Raghu Vansh, live like a Human Being, and eventually, suffer like every Human being does.
You all know the character of Manthara, and her personality. She disliked Shri Ram even before he was proclaimed future King, and she tried many times to change Mata Kaikeyi's affection for him, but why was it that only at the time Dasharath proclaimed shri Ram as future king, Manthara's words poisoned Kaikeyi's mind? Why not any time else? Why was it that every other time, Kaikeyi told Manthara to hold her tongue and stop talking nonsense?
That's because it was fate that Shri Ram had to kill Ravan, and he couldn't exactly kill him while ruling the kingdom of Ayodhya, could he? Shiva's vahana Nandi once cursed Ravan that Vanaras would have a hand in killing him, so that's why Sugriv's vanaras became Shri Ram's army instead of the soldiers of Ayodhya.
I also once heard a story from my grandmother and mother that after Dasharath proclaimed his son as future Raja, Goddess Saraswati came down from heaven in a very miniscule form (smaller than an atom) and sat on the tongue of Manthara, so that whatever Manthara said would influence Kaikeyi instead of being ignored like all the other times. Only after Bharat disowned his mother did the maya of Goddess Saraswati disappear, leaving Kaikeyi very confused with her actions and wondering why she had done what she did.
And I don't blame Kaikeyi at all for what happened, because it says in the Ramayan that she loved Ram equal to, if not better than, her own son Bharat. It was the maya of Goddess Saraswati that influenced her actions, because Shri Ram had to be exiled, and Ravan had to be killed.
But if you think about the Ramayana in a story kind of way, and relate it to these days, I guess most people would blame Manthara and Kaikeyi more than fate. Even then, I would blame Manthara more than Kaikeyi.
I think this issue is very interesting to think about. That's why I created this post. I wanted to read everyone's opinions about who was to blame for the exile.