Possibilities:-
1. Burnham misinterpreted the data
2. Ten years is a long time politically for an empire, anything can happen including a few social reforms
3. This is not the Prime Mirror universe
4. You assume every human agrees with how the empire operates
Burnham misinterpreting the data is an easy answer, but it's hard to dismiss the writers' intent by saying the character is wrong. So far, no evidence other than TOS that Burnham was wrong.
Number 2 doesn't seem likely since by the time we see Mirror Mirror, the Empire is going quite strong, and that episode set the template for this one. Plus, Spock would be on the Enterprise at this point, if his history matches Spock Prime, which it should given that they both end up on the Enterprise.
Number 3 is a fair explanation, and a very good one, though again it comes down to writers' intent. I think both universe from Discovery would be better if they were not what we know.
Number 4 I don't feel is relevant, because even if that were the case, that really wouldn't effect Spock because to be on the Enterprise, and rise to power, you kind of have to buy into it.
Your main issue here is that you're trying to apply logic to the mirror universe. If you stop, the issue disappears.
More like I'm trying to apply consistency to the writing and what we know, based on writers' intent. I obviously don't like the idea that they are visiting the Mirror Universe before Kirk and crew encounter it, which they should first. This show would be in much better shape writing wise if it were set between the launching of the Enterprise B and TNG.
It could then follow up TOS stories without revisionist history.
But here, this is the writers' fault. Humans and Vulcans should be allies in this empire.
Regarding the Temporal Prime Directive, I don't know if something like that would apply. If I know you are going to die, it's not wrong to warn you. As from Discovery's perspective, the Defiant's fate is in their future, not their past, I would not think there are any time travel rules being violated.
It's no different than getting intel that a bunch of Klingons are going to kill the Federation President and stopping it.