The most important rule is to match the stakes with the promises you've made to the reader
You most emphatically do not have to constantly raise the stakes to make a compelling story. As your instincts suggest, switching to more personal stakes can create the same amount of reader investment as higher stakes would have.
For example, Star Wars opened with the fate of the entire rebellion in the balance. But in The Empire Strikes Back, the stakes are narrowed to only focus on the fate of the handful of main characters. But it was no less engaging because of these narrow stakes, because of the emotional connection to these characters.
In fact, I would argue that constantly increasing the stakes is a trap that should be avoided - when the world is always doomed, the stakes stop having the impact that they should, and every new over-the-top threat starts to feel ridiculous rather than dramatic. (You can see this somewhat in the Star Wars Legends, which is filled with dozens of superweapons
The important thing is to make sure that the readers are properly invested in the stakes that you want them to focus on. If your readers feel that they have been promised an epic confrontation between humanity and the forces of evil, then sidelining that epic confrontation to focus on an inter-human conflict will be disappointing. On the other hand, if the readers are expecting a highly emotional, deeply personal struggle for the throne, then sidelining that conflict by threatening the extermination of humanity will be just as upsetting.
If Game of Thrones fans are upset about the final season, then that indicates a failure on the part of the show writers to properly direct their viewer's interest to the conflict that the show writers wished to focus on. (There will always be some readers/viewers who fixate on the 'wrong' conflict, that's to be expected, and is nobody's fault. But if the majority of viewers are reacting poorly in the same way, that's a failure on the part of the show creators.) It says nothing about the intrinsic value of each of the two conflicts.
Television doesn't do denouement conflicts very well
The 'Season Finale' is an important part of the television structure. There is an expectation that the final episode of a season will be the culmination of the themes and conflicts that the season has been setting up.
In the Lord of the Rings novels, after Sauron has been defeated and the king is crowned, the Hobbits return home and have to deal with the Scouring the of the Shire. This works excellently in the book because the book is structured as a journey, which ends with a return to home which contrasts who the journeying characters were at the beginning of the journey with who they are now. The movies, on the other hand, cut out the Scouring of the Shire, because The Return of the King is told as a war story. It ends with the victory over the enemy, and doesn't have space for an extended denouement about the return journey.
Were The Lord of the Rings a TV show, it would likely have also cut the Scouring the Shire, because as much as the Scouring of the Shire is an important part of the book, it isn't the main conflict, and the Season Finale needs to conclude the primary conflict of the season. And after that conflict is resolved, there's not enough room in the episode for another, lesser conflict.
EDIT: Further thoughts comparing Game of Thrones to Lord of the Rings
Pace noted in the comments that GRRM has said that he was aiming for a similar tone in his ending to The Lord of the Rings. If the TV show ending is truly seeking after Tolkien's ending, I think that it missed the mark. Even though superficially they both seem to have a major climax followed by a lesser conflict, they are actually more different than they are similar.
The Lord of the Rings is structurally symmetric. It opens with Bilbo realizing that he no longer belongs in the Shire and leaving. Then Frodo and his friends flee the Shire to escape danger they do not understand. They meet Aragorn and learn of his status as King in Exile, and then finally at the Council of Elrond determine that they must destroy the Ring.
At the end of the novel, the Ring is destroyed, Aragorn is crowned King, the Hobbits return home (handily dispatching danger that they are now thoroughly in command of), and finally Frodo comes to understand that he no longer belongs in the Shire, and departs.
The Game of Thrones does not have this same symmetry. It opens with an attack by a White Walker, followed by a demonstration of Ned's honor, and then the Starks enter the Game of Thrones.
At the end of the series,
The White Walkers are defeated, Jon demonstrates his honor (by choosing duty over love), and finally the Game of Thrones is decided (when Bran is crowned king).
Rather than the first in, last out structure of Lord of the Rings, Game of Thrones is thematically first in, first out. I personally believe that the failure to mirror the introduction of the themes and conflicts is part of what's causing dissonance among the fans of the show.