Yeah like I said it would be somewhat difficult to include in there. I agree that he is much better served as an unexplained deity that supernaturally knows how the story ends for the main character. But I feel it wouldn’t hurt for him to know even more about Kylo (his heritage).

If I was to modify what you have written it would be like the following (changes in bold):

The creature spoke to Kylo in a soft, distant voice, directly into his head. “I am the Eye of Webbish-Bog. I know what you seek, heir of Vader.”
“You will give it to me,” Kylo ordered with a wave of his hand.
“No need for that,” the Eye said plainly. “Do you really think your grandfather would have left it in the stewardship of one who could be swayed so easily by a trick of the Force?”
No, he supposed not.
“I must warn you,” the Eye continued. “If you proceed down this path, you will embrace your destiny. You will become what you were meant to be.”
At these words, ambition burned in Kylo. He had killed the past and watched it die, time and time again, only for it return to haunt him. This creature knew it.
“I know what awaits,” he replied. “Where the contents of that chest will take me.”
“Oh yes,” said the Eye. “You know. You certainly know!”
It cocked its head, making an eerie squealing noise. It took a moment for Kylo to realize the creature was laughing at him.
“Yes, you know,” it said. “Finish what he started!”
Kylo’s lightsaber burst to life as he raised it above his head, but the Eye of Webbish-Bog had already sunk beneath the water, leaving only an uncannily still surface and the echo of its unpleasant squealing laugh.

The final line has a double meaning. Vader wanted to rule the galaxy, but also his original intentions were to save the one he cares about from dying (which is what Ben ends up doing with Rey).