Beautiful, this analysis. And how you have described relationships using the turn of seasons is probably something that the writer wants to convey to the viewers. (Sadly, not all in MDL could realize this.)
My take on this is that watching this drama from a wider perspective and not focusing on the small flawed details will make you appreciate the story more.
The title "꼭두의 계절" has been translated as "Kkokdu: Season of Deity" for its title English. But it could simply be read as "Kkokdu's Gye Jeol" which, for me, simply means that from the start until the end, whether it be "Oh Hyun", "Do Jin Woo" or "Kkokdu", he will always have "Seol Hee" or "Gye Jeol" in his heart (and it doesn't even matter if his mind remembers her face or not). And the fate that awaits them every reincarnation is not death but the love that draws them to each other. I am not talking about the boy meets girl and they fall in love then get married kind of love. The love I am talking about here is the love that you have clearly described above.
The kind of love that withstands the test of time. Or in this case, the test of seasons. The kind of love that unfolds (not just happens in a snap), that grows (not immediately grand in appearance) and remains (does not easily gives up and leaves).
To what can we compare that love? I could describe it as a tree.
From a sapling a tree grows. And when the tree grows into its full form, it withstands every season: summer, fall and winter; and appears like a new creation when spring comes. All the while, it seems like the love between Oh Hyun and Seol Hee is a cycle of summer, fall and winter, but not spring. How come? Probably because the love inside Oh Hyun became stubborn rage against the deity and the extreme coldness and extreme heat of the underworld made it seem like spring would never come. But his 99 years on earth is not simply an added punishment, it also serves like a bridge that could either lead him to another cycle of summer, fall and winter or to a new spring which comes from a faithful and courageous love.
"I love you."
Seol Hee/Gye Jeol bravely saying these words despite knowing that Kkokdu might die (assuming that the "peace" that "will come" means death) was spoken from a heart that loved from the beginning and a heart that is willing to love until the end.
Oh Hyun died in front of a tree in full bloom.
Oh Hyun became Kkokdu and lived in the underworld beside a tree that gradually dried up.
Will we see that tree filled with life once again? Only one episode left and we'll find out.
(I feel like I just wrote gibberish. I hope it somehow makes sense. Haha.)