Jon Snow might not have killed the Night King but without him rallying his forces, siding with Daenerys etc then the climatic battle wouldn;t have happened
You are right, only in that the Night King and his army would still be stuck north of the wall twiddling their thumbs for all of eternity, so yeah there wouldn't have been a climatic battle.
If Jon Snow hadn't been brought back to life here is what would have happened:
Allister Thorne and Janos Slynt would of taken over rule of the Night's Watch, probably Allister Thorne as lord commander. They would of petitioned the kingdoms for men to help them round up all these wildlings that Jon Snow let loose south of the wall. The southern kingdoms would have ignored them, though the North would have assisted. Ramsey Bolton's hold on the north was very tenuous, he would have had to help the night's watch just to earn cred with the northern lords. They would of had some skirmishes and battles and probably completely ignored anything happening north of the wall.
In the South Dany would have arrived and took dragonstone, without the doe-eyed ramblings of Jon Snow "yes ma'am"ing her or cautioning her to take it easy she would of just sieged King's landing immediately while she still had the entirety of the Reach and the old iron islands fleet combined with her 40k Dothraki 10k unsullied and 3 full grown dragons, she would have taken the city damn near without a single loss and merc'd Cersei without any consequence. Her unstoppable massive army with 3 winged nukes would have easily dispatched what remained of any lannister forces and then destroyed euron's fleet and turned over control of the Iron islands to a grateful Yara (name in the show) Greyjoy.
without any deaths she would of never been some mad queen, just been her same old mediocre ruler, not great, not terrible. She would listen to council sometimes and sometimes she would make dumbass decisions.
Eventually the issues in the north with the wildlings would resolve one way or another and someone from atop the wall would notice that there was an army of 100k+ undead + about 100 white walkers sitting at the bottom of the wall, unable to cross it, just standing there holding their dicks.
Eventually after enough pleading the queen would of flown north, been completely confused on why her dragons refused to fly near the wall, it's almost like it was some sort of magical fucking barrier that prevented them from crossing it, just like it was a magical barrier preventing the army of the dead from crossing south of it. Perplexed that she couldn't fly north of the wall, she would of just rode up the elevator and saw with her own eyes this massive army of undead standing at the base of the wall, holding their rotting dicks. Luckily none of their number are superman level olympic javelin tossers so they would be in no danger 800 feet straight up at the top of the wall.
She would of shrugged and left, and eventually advisers like Tyrion would arrive to assess the situation. He would of then ordered wagons upon wagons upon wagons of pitch and tar shipped to the wall, the trebutchets and other siege weapons atop the wall rebuilt, and they would of simply covered the entire army of the undead with pitch and then set them on fire.
This would of burned to a crisp everything except for the Night King who was somehow magically immune to fire. The other walkers would have died tho, and all the dead would be ash and unable to be raised again.
Then they would of noticed that the sole survivor, this Night King all alone, still was unable to cross the wall. They could even open the gates and he was incapable of crossing under it, it was a fucking magical barrier all along, who would of thunk it? it's not like every single story and myth and lore from the last 8k years described it as a magical barrier that prevented the others from crossing was actually true?
So then Tyrion would of ordered the best 40 or so fighters in the realm to report to the wall, equipped with valyrian steel weapons, they would line up and just marched out of the tunnel side by side and then play whack a mole on the Night king until eventually one of them managed to land a single blow (ie instantly) which instantly kills him.
