Do vacuum tube test values reflect sound quality?

"A mess" meaning there would have to be at least one or two more tubes, preferably of high transconductance to use as CFs. But could you then delete one of the 12AX7s and end up with the same number of tubes?

While I recognize the importance of the 7C in audio history and its high sonic quality for its day (and its collector value), I don't think it measures up well against any. of the best phonolinepreamps of the past 25 years purely in terms of sound quality.  Not that I can prove that the 12AX7 it uses as a CF is an audible weak link. More likely the power supply and parts quality hold it back, in the sense that we have much better capacitors and resistors of all kinds available these days. Also, the one and only time I ever saw a schematic, it seemed overly complex, but I cannot recall specifics.

In a well designed circuit, tubes will last quite a long time, certainly many years, sometimes more than a decade. One of the few good uses for a tube tester (and I specify a tester that measures transconductance in units rather than as “good” or “bad”) is to record the values for Gm when new and then retest the tubes periodically if you want to ameliorate your anxieties about tube aging. Hickok made the best testers and they all can measure Gm, for example. Should one prefer SS because of tubes aging? With SS you get sudden total failures where the discrete transistor or integrated circuit that failed is no longer available. Nor is there anyone left with expertise to do the repair. With tubes you just plug in a new one.

For a cathode follower you WANT high transconductance. Tubes like the 12AX7, used by many companies as a cathode follower, are not a good choice. They do it because there are billions of 12AX7s in the world, and more are being made in Eastern Europe. The CF changes voltage to current. 12AX7s are puny for current. 

"Triodes that are unbalanced sound just as good as long as one side tests strong." That’s true, IF the gain circuit is single-ended as opposed to balanced, and if the section with high transconductance is the one you are using. This is often how tubes become unbalanced; they have been used in a single-ended circuit which only requires one of the two sections where the other section (of a dual section triode) is not doing anything. Some canny manufacturers of SE preamps build their units so that one channel uses section 1 of a dual triode, and the other channel uses only section 2 of the same type dual triode. (Two tubes, one for each channel.) That way, when the tubes become worn, you can swap them from one side to the other of the stereo circuit and in essence you have new tubes. Balanced sections is much more important for a balanced circuit. Also, transconductance is not necessarily a correlate of "gain". Depends on the circuit and the tube. For example, a 12AX7 has highest gain among typical dual triodes, It is commonly used at the input of MM phono stages, because it has highest mu. Mu is equal to Gm*RP (transconductance X plate resistance), and the 12AX7 has very high plate resistance but relatively low Gm. Mu is a better correlate of gain. Gm is more a correlate of how the tube handles current.

The only thing a tube tester can tell you is whether a tube is worn out or nearly so. That you can determine by comparing the manufacturer data to the tube tester result. But the red flag is transconductance.  As tubes wear, transconductance goes down, for any given bias and plate voltage. Consult an RCA tube manual or any of several websites to download the pdf file for the tube under investigation.  These are very old data from manufacturers that no longer exist or no longer make tubes, but the numbers are still valid ways to evaluate tube condition.  If the measured Gm (transconductance) is below the range of values given for a given bias and plate voltage, then the tube is worn or wearing out.  Yet, as Larry said, that tube may still sound "good".  Better test is to replace such a tube with a new one and then listen. If you hear improvement in terms of a livelier sound or better treble and bass response or a sense of better gain character, dump the worn tube or save it for a spare.

The Chinese-made fake ECC81/12AT7 often are labeled as "801S", just to make them look even more desirable.  On the other hand, these fakes often sound fantastic.  I compared about 6 different NOS 12AT7s in the input stage of my Atmasphere amps, many years ago. Other than the Chinese made 12AT7s, all the other tubes were true Euro or US NOS brands such as Amperex, Sylvania, Mullard, GE, etc.  The Chinese tubes sounded by far the best.   I bought them from Billington in the UK, and they were openly sold as Chinese tubes under the label "Billington Gold" (they had gold pins); there was no attempt to mislabel them as TFK or anything like that.  The downside of those Chinese 12AT7s was only that they had a short lifespan compared to true NOS tubes. I used them anyway for a while.

These days, for the exotic types and brands you mention (Siemens and TFK), I’d hazard a guess that there are more fakes than real ones on the market. I’d stay away from eBay as a source for such tubes. Most reputable dealers don’t pretend to have such tubes to sell. I assume you know that “ECC801”indicates a selected ECC81, even more rare than the run of the mill version of Siemens or TFK. What few exotic small signal triodes that I own I bought in the 1970s, so I know they are for real at least. However, maybe you are an expert in examining these tubes to prove to yourself that they are what they say they are. I certainly am not, even after 45 years in this hobby. To the question you did ask, yes I think 3000 and 2750 are sufficiently well matched sections.

There is no test that correlates with sound quality except you want transconductance to be well within the good range and the two sections of a 6SN7 or any dual triode to be about equal in transconductance, but for two tubes that test well, no measurement is going to tell you that one sounds better than the other. Furthermore, with octal triodes you need to be aware of microphonics. Some are very sensitive and THAT can color the sound.