The problem is not "a" backstory. The problem is nothing but backstory.
Explain what you mean by "nothing but backstory".
When she was introduced, she had the backstory of Tom Paris, Seven of Nine and Worf all combined.
Ah in other words, Burnham was the show's main protagonist. Yeah, in Trek shows and others, that sometimes means that character gets most of the attention. I think this is once again, a case of forgetting that in DSC, the captain isn't the main character.
So perhaps seeing what would usually be a secondary character (the ship's first officer), get most of the backstory, the kind of diverse background usually reserved for the captain character, makes it seem like it's
too many backstories. I'm going to start calling this "Lord Garth Syndrome", since he was the first to diagnose it.

BTW, you didn't explain why "too many" backstories is a problem.
What happened during the run of season 1 was that the creators simply dropped half of her backstory - her acting like a Vulcan - gone, her being traumatized by the klingon attack - almost gone (except for that horrible reappereance in the finale)
They dropped NONE of Burnham's backstory. Which parts do you think were dropped?
Burnham's "trauma" from the Klingon attacks had been resolved at some point prior to her joining the Disco crew. It was not a factor during the season. Also, I think she acted as Vulcan-like at the end of the season as she did from when we see her and Georgiou together at the start of the pilot. However, spending so much time with humans (again), no doubt had an affect on her as we've seen it have on half Vulcans and full Vulcans like, Spock, T'Pol, and Tuvok.
It's not that during the season we got to know her character better. It's that they needed the entire season for her to actually become a character in the first place.
Well, IMO, as to when Burnham became a fully fleshed out, fully realized character, I would only disagree with you as to when it happened. I saw the character this way by the end of the third episode. That it didn't happen for you until season's end is not that big a deal. At least it happened in the first season. Some shows take much longer to get even their main character to this point.