In 1903 the Canada-Alaska border dispute was decided by a tribunal of 3 American members, 2 Canadian members, and 1 British member. The British member sided with the Americans with the stated goal of improving relations with the USA. Was this a needless concession or a gift vital to maintaining friendly relations with the USA?
In 1903, the President of the United States was one Theodore "Teddy" Roosevelt, who was arguably the most pro-British President in modern American history. He famously made a remark that America and the British Empire together could "whip" the rest of the world. Had he been President in 1914, he certainly would have brought the U.S. into World War I on the British side. His son, a Brigadier General, fought at Normandy in World War II.
In 1905, Roosevelt negotiated the Peace of Portsmouth in the Russo-Japanese war in favor of Britain's ally, Japan. In 1906, Roosevelt secretly supported Britain's ally, France against Germany regarding Morocco, while pretending to be an "honest broker." (But he may have pushed back World War I by a decade.)
Basically, it made sense for Britain to cater to Teddy Roosevelt. Whether it would, or should have done so for another President is open to question.