The Korok Trials are not the same as the Trial of the Sword.

The Korok Trials are what you were doing: Trial of the Second Sight, The Lost Pilgrimage and the Test of Wood. The only things you get from those is one shrine each and then a set of 3 Big Hearty Truffles from the elder (the dude with the mushrooms on his head).

The Trial of the Sword also takes place in Korok Forest, but it is DLC content where you have to put the Master Sword back into the pedestral and then clear multiple different "stages" of a You-VS-Monsters setup without your equipment. It's like Eventide Island, but bigger.

...and technically it doesn't "upgrade" the sword either. While you're in the presence of Malice, the Master Sword starts to glow and gets enhanced durability and 60 attack power - this already happens in the base game. The only thing the Sword Trial does is make the Master Sword have this "powered up" state permanently even while you're not near Malice, but it neither powers up again nor will it become stronger than 60 ATT, ever. Personally I don't find it that useful TBH since the only times I ever actually use the Master Sword is durig boss battles (where it is always powered up anyway, so clearing the Sword Trials would not make a difference) and for mining ore (since it recharges anyway in both cases).

But to answer your actual question: You can permanently fuck up the "Lost Pilgrimage" quest. If you skip talking to Oaki (the little Korok dude you had to follow) and head to the shrine right after completing the trial, you can lock yourself out of actually finishing the quest, and since the shrine is unlocked then, the game won't let you restart it either. This however isn't that important ... annoying, but not important. There is no reward for finishig all quests, so the only thing you're missing out on are the 3 Big Hearty Truffles the elder would have given you, and you can find that stuff elsewhere.