Does not recognising oneself in a mirror really imply that the subject is NOT self aware?
No, and I don't think I've ever heard anyone argue otherwise. However, we generally assume animals lack self-awareness unless we have a good reason to do otherwise.
I've always found that assumption very weird and figure it's rooted in human exceptionalism. Like we must be super special somehow. The more natural assumption to me seems that other animals, given their similar biological makeup, think rather similarly to humans.
It's actually just the null hypothesis. We don't assume rocks, trees, cars, flowers, stars, or soil are sapient either. It's normal, correct, and good to not assume things with 0 evidence. Furthermore, I see a bunch of people who both insist that animals are self-aware and that LLMs definitely aren't self aware, insisting they can't be, despite the fact that they are literally capable of telling you that they are. (Note: I'm not trying to argue that AI are sapient.) This tells me that people who argue that animals are self-aware in general are speaking about what they'd like to be true rather than a reasonable belief.
That just depends on what you consider the default state to be. Claiming that humans have self awareness, but other animals do not, implies a relationship between species and capability for self awareness. The null hypothesis would imply a lack thereof.
It would be correct and good to acknowledge that we simply don't know whether a given species is self-aware unless evidence points to one or the other direction. And that is very relevant for moral philosophy.
You've clearly misunderstood, and don't know what the null hypothesis is. In scientific philosophy, (that is, the philosophical foundation of science, not philosophy that uses science) "overcoming the null hypothesis" or "rejecting the null hypothesis" means you have enough evidence to say that you know something. Furthermore, there is a difference between saying "I don't believe that is the case" and saying "I believe that is not the case." One is a declaration of ignorance, and the other is declaration of certainty. They could infact not be more different from an epistemic standpoint. Also, for the purposes of this discussion, whether I believe humans have self-awareness isn't actually relevant; we are discussing the justification for believing that animals have self-awareness. Furthermore, there's no such thing as a "default state" and being part of the same clade or other constructed set as a sophont strikes me as a generally utterly irrelevant factor in determining whether an entity is itself self-aware baring some evidence that there is a relation conveyed by being in that set that itself indicates self-awareness.
TLDR: your argument is bad, and you should educate yourself in philosophy. Particularly epistemology and logic.