Well mostly the flaw is people assigning the test abilities it was never intended. Like testing intelligence. Turing outright as first thing in the paper presenting "imitation game" noted moving away from testing intelligence, since he didn't know to do that. Even on the realm of "testing intelligent kind of behavior" well more like human like behavior and human being here proxy for intelligent, it was mostly an academic research idea. Not a concrete test meant to be some milestone.

If the meaning of the words ‘machine’ and ‘think’ are to be found by examining how they are commonly useit is difficult to escape the conclusion that the meaning and the answer to the question, ‘Can machines think?’ is to be sought in a statistical survey such as a Gallup poll. But this is absurd. Instead of attempting such a definition I shall replace the question by another, which is closely related to it and is expressed in relatively unambiguous words.

Turing wanted a way to step away from stuff like "thinking" and "intelligence" directly and then proposed "imitation game" mostly to the rest of the academia as way to develop computer systemics more towards "intelligent behavior". It was mostly like "hey we need some goal to have as a goal to have something to move towards with these intelligence things. This isn't intelligence, but it might be usefull goal or tool for development work". Since without some goal/project/aim to have project don't advance. So it was "how about we try to develop a thing, that can beat this imitation game. Wouldn't that be good stepping stone. Then we can move to the actual serious stuff. Just an idea".

However since this academic "thinking out aloud spitballing ideas" was uttered by the Alan Turing, it became the Turing Test and everyone started taking it way too seriously. Specially outside academia. Who yes did play the imitation game with their programs as it was intended as research and development tool.

exemplified by for example this little exerpt of "not trying to do anything too complete and ground breaking here":

In any case there is no intention to investigate here the theory of the game, and it will be assumed that the best strategy is to try to provide answers that would naturally be given by a man

It is pretty literally "I had a thought". Turin makes no claims of machine beating the game having any significance other than "machine beat this game I came up with, neat". There is no argument of if machine beats imitation game, then X or then it means Y is reached.

Rest of the paper is actually about objections to the core idea of "it could ever be possible for machine to think" and even as such said imitation game is kinda lead in or introduction to Turing's treatise various objections of various "it would be impossible for machine to think" arguments. Starting with theological argument of "only human soul can think. Hence no animal or machine can think." .... since it was 1950's.