Hmm, alright, that even makes a certain amount of sense. However, it is clear that such behavior is extremely problematic. "Generic answer" may sound kind of benign, and in my original example the generic answer is indeed correct most of the time, but it is clear that it will be possible to construct expressions for which Mathematica will give "generic" answers that are almost always wrong.
More importantly, a computational system that even just sometimes produces wrong results is already fatally flawed. In numerical computation, nobody would accept a floating point unit that sometimes gives the wrong answer, even if only very rarely (see the infamous FDIV bug in the first-generation of Intel Pentium processors). Common judgment of such devices is that they are completely useless. Likewise, if I want to trust Mathematica's answer in a potentially complex set of interdependent calculations that I may not be able to easily check "by hand", I have to be convinced that the results that Mathematica generates are always correct. If that is not the case, how can I trust any result that Mathematica is giving me?
To be very specific, I find the statement that "If the Wolfram Language did not automatically replace 0/x by 0, then few symbolic computations would get very far" to be quite a lame excuse. Why not demand that the user specify x such that an unambiguous answer is possible, or else produce an error message? So, calculations become impractical if we insist that they give correct answers? And our choice is then that we'll just give wrong answers instead? Are we saying that, "well, alright, we're not sure we're giving you the right answer, but at least it's an answer"?
So, no, I do disagree that "The overall goal of symbolic computation is typically to get formulas that are valid for many possible values of the variables that appear in them" (in parentheses I will note the weasel-wording through the insertion of a "typically" in the sentence above). As for myself, I want a correct answer, period, not an answer that is often correct. To be provocative, I will go so far as to say that, just as in my example of numerical computation, a symbolic system that sometimes gives the wrong answer is useless. Feel free to discuss...
P.S.: As a consequence of what I am saying above, in my opinion, the solution of x=0 to the equation a x=0 is wrong; Mathematica should have given the solution as x=0/a. In contrast, the solution x=-b/a that Mathematica generates for the equation a x+b=0 is correct.