I cannot perceive the awkwardness in "That's your only friend that I've ever met".
Good. In informal modern English, there isn't any.
The placement of the genitives and clauses here is no more wrong than the admonisher using that instead of whom in their own 'model' sentence. You might then ask
Why do they think it's impossible?
It's rough going for ELLs but the best treatment of some people's tendency to beclown themselves with overstatements like your quote was David Foster Wallace's "Tense Present" for the April 2001 Harper's. In this particular case, Google won't show the text you quoted but they have it scanned and indexed. They lay credit/blame for your passage at the doorstep of the 1992 edition of the American Heritage Dictionary edited by Anne H. Soukhanov. Her own justification for being a SNOOT despite knowing better would be along the lines of
... you're basically studying a foreign dialect. This dialect is called Standard Written English... There are some otherwise smart English profs who aren't very aware that there are real dialects of English other than SWE, so when they're reading your papers they'll put, like, "Incorrect conjugation" or "Comma needed" instead of "SWE conjugates this verb differently" or "SWE calls for a comma here." That's the good news—it's not that you're a bad writer, it's that you haven't learned the special rules of the dialect they want you to write in.
Still, even though Ms Soukhanov is an intelligent woman, in this case she isn't defending good grammar or logic against the careless hordes. She is choosing to understand the sentence in a particularly unhelpful way, then declaring that way completely wrong, and then ignoring the other ways to understand the expression that make it perfectly clear to native speakers.
She is taking the sentence to be the conjunction of the ideas That is your only friend and also that is the only one of your friends whom I've met. Those two ideas are mutually exclusive. Subsequent editions of the AHD and their Book of English Usage have walked back her claim a little. They now just say it's ungrammatical or that some people will find it ungrammatical. Really, of course, the way she (and some of the posters here) are choosing to misread the sentence is itself a mistake that tries to reuse the only for two separate meanings at the same time.
Like you already know, that sentence actually just means That is the only one of your friends whom I have met and there's no problem... unless you're doing a test with subjective grading and need to keep the scorer happy. In that case, the safer thing to do is rephrase.
(In Ms Soukhanov's defense, she was somewhat distracted. In this passage, she was trying to defend a good informal usage against people even more proscriptivist than she is, who want to pretend double genitives are never appropriate. She then tried to draw her own line to show she wasn't on the side of complete descriptivist chaos. I agree with you that her example wasn't well considered. Even Astralbee above thinks the sentence is a possible idea; they just dislike the sloppiness of using a restrictive clause on someone just described as totally unique.)