There’s a few things to consider, with this:
1) Burnham has known the Federation all her life, and lives her own life by the Federation code, so to speak. It’s existed the entire time she’s been alive, so the idea that it may no longer exist probably hasn’t entered her mind.
2) The plan to travel forward in time was crafted during a very time-sensitive period. We didn’t see them discuss anything about what the future may be like when they get there, if I recall? So, she simply hasn’t had time to consider the very likely possibility that the Federation would not exist once she completed her journey through time.
3) For Burnham, this all happened in the blink of an eye. One minute, she was in the 2250’s, the Federation not yet even anywhere near its peak. The next, she’s in a strange new world, and she’s told there’s no Federation (mostly). All that, in the same day. It’s a lot to digest.
Hell, if she had time to stop and think about everything, she’d probably have soon also faced the harsh reality (to her) that the moment she arrived in 3188, Spock was long dead, from her perspective.
I think that’s all a lot to digest, at the best of times. All in the same day? Nigh on impossible.