Defeat by the USA on Wednesday had left Britain needing to beat the Dutch and probably Kazakhstan as well but Sunday's contest will now be a meaningless match after Andy Murray and Joe Salisbury again came out on the wrong side of a dramatic deciding doubles rubber.
Instead it is the Dutch, with their small band of orange-clad supporters, and the Americans who can celebrate their progress to the last-eight shootout in Malaga in November after two wins each from their first two matches.
Britain had to pick themselves up after they were edged out by the US very late on Wednesday night, and they made the perfect start when Dan Evans defeated Tallon Griekspoor 6-4 6-4.
But Cameron Norrie then produced a poor display to go down 6-4 6-2 to Botic Van De Zandschulp, and Murray and Salisbury were beaten 7-6 (0) 6-7 (6) 6-3 by Wesley Koolhof and Matwe Middelkoop despite the best efforts of the Emirates Stadium crowd to will on their home hero in his 50th Davis Cup rubber.
Murray was again given the nod over Neal Skupski, who reached the men's doubles final with Koolhof at the US Open last week, and he and Salisbury made a great start with a break of the Middelkoop serve in the opening game.
But they could not hold onto the advantage and the Dutch pair played an inspired first tie-break. The second tie-break was full of drama, with Britain saving a match point and then taking their first set point thanks to a helpful net cord.
Players, team and supporters celebrated joyously but they still had to win the decider and, after missing chances at 3-3, Salisbury dropped serve in the next game, leaving Koolhof to serve out the victory.
Captain Leon Smith has never had greater strength in depth at his disposal but with that come selection dilemmas and he will need to reflect on what is the most disappointing week of his long and highly successful captaincy, particularly given Britain's home advantage.
They are due to host a group stage for the next four years, but only if they qualify among the best 16 nations, which is likely to mean winning a play-off early in 2023 for they surely cannot expect another wild card.
Norrie had scrambled his way to victory over Taylor Fritz on Wednesday but this time was unable to recover from a bad start against a player he beat easily in Montreal last month.
The world No8 has struggled to find his game on the very slow Glasgow court and was second best throughout against Van de Zandschulp, who served 13 aces.
Norrie said: “He was better than me. I just gifted him two service games. He didn't really have to do much to break me in the end. That's not a player that's of my ranking, and that's me on a really bad day.
“I was really impressed with Botic and his level. Especially the last time we played, I played a very clean match on my serve and gave him absolutely nothing. He did that today to me.”
Evans, who lost a high-quality clash with the US's Tommy Paul, did not have things all his own way against Griekspoor but played a very solid match, breaking serve once in each set.
The British No2 said: “It was a good match. I played well. More of a match of just focus to get through really, to get a win.
“I knew I was obviously playing better than him, the higher-ranked player in the match. I just had to put my game on the court, make sure I focused at the right moments and put pressure on him.”