I've been trying to research this a little too, to understand it better. Bear in mind, I've not fully finished the series, I'm mid season 3, so anything we might learn after this, I don't know yet. (Nor have I found much info outside of this so far either. )
It seems to me though, that the reason why Meliodas destroyed the entire city in rage after living through 106 deaths is quite understandable. He's lived, loved and lost the love of his life for 3000 years. On top of this, he's come no closer to fulfilling his promise to her. As we see from the few other reincarnations we see, this Liz is one of only a few he's made a full life with. And might also be one of the few reincarnations he let himself go with.
This would perhaps lead to him falling into darkness, being consumed by rage, both as a reaction to why and how Liz died, but also as a reaction to his own lack of being able to stop it from happening again.
I also feel like the 106th Elizabeth is supposed to be special in some way not yet explained; she's the only reincarnation thus far that's had clear differences from all her other reincarnations. You can explain clothes and hairstyle, even personality to some extent being different each time, based on where and how she's raised but as the first reincarnation explained too.. Part of their curse is for her to be reincarnated in the image of the original Elizabeth. Eyes, hair color, facial feature, even personality traits are meant to be similar, to ensure Meliodas not only meet her but recognize her and falls in love with her. As well as Elizabeth falling in love with Meliodas. That's their curse.
As a last point, Meliodas retells of a few different reincarnations. Each of them tells us about Elizabeth being destined to die within 3 days of regaining her memories, but as perhaps an added trigger to his anger, we learn of only one incident where Elizabeth never regained her memory and thus lived a long life, dying peacefully of old age.
This is speculations on my part, but.. If Meliodas knew he was destined to fall in love with every reincarnation, and he learned Elizabeth can live a full life, and die peacefully.. Liz dying in such a painful and violent way without having any memory of her past lives may have helped drive him into despair. Not only had he not been able to lift the curse, he couldn't even protect her from a brutal end even when she hadn't regained her memories.