I can only offer speculation as I don't know of any official source that explains Bo-Katan's motives during the events in The Mandalorian. In fact, her story hasn't been fully told yet. There might be parts of her backstory that haven't been published yet and that shed more light on her reaction. We don't even know how she lost the Darksaber in the first place, for example.
But based on what we know, I would like to offer a slightly different take than the other answers. Based on Bo-Katan's history my guess is that tradition and appearances aside, this is actually a personal issue for her. Because Bo-Katan never won the saber in the traditional sense in the first place.
Depending on how closely you interpret the rule of having to win the saber from its previous owner, you might say that Maul was the last legitimate owner and everyone after him was not. He challenged Pre Viszla to single combat and claimed both the Darksaber and the Mandalorian throne after defeating and killing him. Maul kept the blade even after he was ousted from Mandalore and eventually kept it on Dathomir along with other artifacts from his past.
Years later (and to make a long story short) it came into the possession of Sabine Wren and thus found its way back into the hands of the Mandalorians.
Sabine was looking for a strong and respected leader to unite the Mandalorian clans against the empire. She believed Bo-Katan to be that leader and offered her the saber. But Bo-Katan didn't share Sabine's view of herself and refused. After her past mistakes, she doubted herself and her ability to lead. She thought she had failed her people and was unworthy to lead them.
Sabine ultimately convinced her that she was the best choice and showed her, that she already had the support of the Mandalorian clans even without the saber. Sabine was convinced that she only found the Darksaber so she could pass it on to Bo-Katan.
Only then Bo-Katan accepted and Sabine simply gave her the saber in front of the assembled Mandalorians. And this is where things get tricky. Everyone was absolutely fine with it even without her earning the Darksaber properly. Tradition aside she was accepted as their leader. Or at least that is the way it's depicted in Star Wars Rebels.
Like I said at the beginning, we don't know the full story. We don't know everything that happened between then and the events in The Mandalorian. It could be that her rule wasn't as unanimously accepted as it appeared. Maybe it was disputed by some from the start. Or maybe her leadership was put into question after the events surrounding the Great Purge and the loss of the Darksaber.
Regardless it seems from prior precedent that bending the rules and simply accepting the saber isn't totally out of the question. The situation might have changed insofar as that she has lost so much support, that the other Mandalorians won't accept such a claim to rulership a second time.
But my guess is, that she needs this more for herself. I think she needs to properly lay claim to the Darksaber this time not just to convince the other Mandalorians to follow her but also to prove to herself that she is a worthy leader. She doubted herself before and it stands to reason that she would do so again after not being able to save her people from the purge and also losing the symbol of rulership.
She needs to do this the right way to prove that the trust in her was not misplaced.