Short version: you're right, there is a serious issue here, and it comes down to the exact rules we use to form terms. You've phrased it in terms of trivialization ("doesn't this reduce the proof of associativity to the reflexivity of "$=$"?"), but it's a more serious problem than that, even: it ultimately will allow us to prove that every (infix) binary operation is associative, which is clearly bonkers$^*$ (see my comments below the OP). So something needs to be resolved here.
Ultimately we want to not consider expressions like "$x+y+z$" to be terms at all (which makes sense, because until we've proved that $+$ is associative there's no reason to expect "$x+y+z$" to unambiguously refer to anything).
So it's not that "$(a+b)+c=a+b+c$" is false, but rather that it's grammatically incorrect.
Unfortunately some presentations gloss over this point in a way which does in fact cause a serious problem. I think the simplest approach is to outlaw infix notation entirely (so that "$x+y$" is a term in the "metalanguage," but not actually in the formal language we use). This has the drawback of being hard to read, but fairly quickly you'll reach the point where you're comfortable using informal infix notation as shorthand for the formal expressions in question.
The syntax I present below is just one option of many:
Given a language $L$ consisting of constant symbols, function symbols (with arities), and relation symbols (with arities), we define the class of $L$-terms as follows:
Every constant symbol in $L$ is a term. (Note that there may be none of these.)
Every variable is a term. (Here variables exist independently of the language $L$; in other words, variables are "logical symbols" like $\forall$, $\wedge$, $($, and $=$.)
This is the key: if $t_1, ..., t_n$ are terms and $f$ is an $n$-ary function symbol in $L$, then $f(t_1, ..., t_n)$ is a term.
No string of symbols is a term unless it is required to be by the rules above.
Note that this syntax does not allow infix symbols: the expression "$x+y$" is grammatically incorrect, and should be written "$+(x, y)$" instead. Note also that "$+$" can only take in as many inputs as its arity: so $+(x, y, z)$ is not a term, since $+$ is $2$-ary. Now the associativity of a binary operation $*$ is expressed as $$\forall x, y, z(*(x, *(y, z))=*(*(x, y), z)),$$ and the issue you raise doesn't crop up here at all since the relevant "ambiguous terms" aren't even well-formed expressions in our syntax.
Incidentally, note that the notation I've described above is redundant: we could write "$ft_1...t_n$" and this would be perfectly unambiguous. But in my opinion this leads to a significant loss of "human readability;" why make your syntax minimal if it's harder to read? You'll pry my unnecessary commas and parentheses from my cold, dead hands!
Now in some sense this can be viewed as dodging the point: what if we do want to allow infix operations? Here again the point is that we need to approach the term-formation rules carefully. My choice would be the following rules (here "$*$" is a binary infix operator):
- If $t_1, t_2$ are terms, then $(t_1*t_2)$ is a term.
Note the addition of parentheses: this prevents (again!) the formation of an "ambiguous term" like $x*y*z$. It does have the drawback of adding "unnecessary parentheses" - e.g. "$x*y$" is not a term, but "$(x*y)$" is - but this winds up not being a problem, only an annoyance.
$^*$Actually, there is a third fix: only allow infix binary operations which are associative! So a language consists of some constant symbols, function symbols, relation symbols, and infix binary operations; and an interpretation of a language has to interpret each infix binary operation symbol as an associative binary function on the domain. This then resolves the difficulties here, since associativity is baked into the choice of notation, and it means we can allow expressions like "$a*b*c$" without causing any problems; however, it seems extremely artificial to me (and goes against mathematical practice as well: we're perfectly happy writing "$x-y$" in the real world, after all), so I wouldn't consider it as anything more than a footnote.