I think this is best answered by seperating the question into two parts.

First, was King George V Euthanized? While you can quibble about the definition of the word based on the king's sharply declining health, he was administered a dose of medication that caused him to die hours (at a minimum) earlier than he might have.
The king had been in his private bedroom for five days complaining of a cold and was drifting in and out of consciousness. On January 20, there had already been a public bulletin to the effect that the king was in his last days. The King's private physician was Lord Dawson of Penn. Lord Dawson wrote in the events in his private diary, which was found after his death and made public in 1986. On January 20, 1936, Lord Dawson wrote in his private diary the following:

At about 11 o'clock it was evident that the last stage might endure for many hours, unknown to the Patient but little comporting with that dignity and serenity which he so richly merited and which demanded a brief final scene. Hours of waiting just for the mechanical end when all that is really life has departed only exhausts the onlookers & keeps them so strained that they cannot avail themselves of the solace of thought, communion or prayer. I therefore decided to determine the end and injected (myself) morphia gr.3/4 and shortly afterwards cocaine gr.1 into the distended jugular vein ... In about 1/4 an hour – breathing quieter – appearance more placid – physical struggle gone.

Second, was this "the reason" Lord Dawson made this decision? We don't truly know that. But we know that he thought about it at least.

The King's family would have opposed any such act had they been consulted. They were not consulted. Separately, Dawson wrote that the timing of the decision was such that the king's death could be announced in the morning Times rather than a less appropriate evening paper. But it's not necessarily clear whether that was any kind of major consideration or whether it simply was one of the things that was in his mind. So the answer is "we don't know," because basically we're speculating about what was inside one man's head at the time he made a very consequential decision.

This information was made public in an article published by Francis Watson in History Today, December 1986. (https://repository.library.georgetown.edu/handle/10822/731931)

Edited: typoes and correcting a date.