tl;dr Yes, in a sense the problem was assuming the harmonic series has a numerical value, and using $\infty - \infty = \log 2$ to conclude $0 = \log 2$.
I sense from the clarifying comments that the real question is not about this specific series, but about a larger issue: If we're given a formal infinite series, when can we assign a real value to the sum in a way that respects usual rules for manipulating series? And so, how do we know we can't assign a numerical value to the sum of the harmonic series?
What does seem clear (unfortunately) is that splitting a divergent formal infinite series into two (or finitely many) "subseries" and manipulating those is not on its own capable of yielding a definition for the sum of a formal series, because we have elementary theorems that at least one subseries diverges, so we learn nothing.
Consider Cesàro convergence: Given a sequence $(a_{k})$, we consider the sequence of arithmetic means $s_{n} = \frac{1}{n} \sum_{k=1}^{n} a_{k}$, and ask whether this converges. If $(s_{n}) \to s$, we say our formal infinite series converges in mean to $s$. This definition gives a unique value to some divergent formal series, e.g. the formal geometric series with ratio $-1$ converges in mean to $1/2$. Cesàro convergence is not a finite splitting: There's an averaging process in which the number of terms being averaged is unbounded.
Similarly, consider regularization by a convergent series of functions, such as the zeta function, or the formal geometric series
$$
\sum_{k=0}^{\infty} x^{k} = \frac{1}{1 - x}.
$$
We can define the value of the formal series on the left to be the rational expression on the right for all real (say) $x \neq 1$. With this definition the formal geometric series with ratio $-1$ "regularizes" to $1/2$. But again we've done something more than algebra: We've extended a single numerical series to a formal power series on an interval of numbers, summed the series where it converges ($|x| < 1$), and then used the fact that the sum makes sense outside $(-1, 1)$.
Do these examples mean "the geometric series with ratio $-1$ really is equal to $1/2$" in some philosophical sense? I'm not going to touch that question, but only point out that mathematically the question is not (in my view) well-posed, because before we ask mathematically, we have to specify how a value is to be assigned to an expression.
For ordinary infinite series, we choose to form a sequence of partial sums and take the limit. It's not (in my view) that "$\sum_{k=0}^{\infty} x^{k} = \frac{1}{1 - x}$ for $|x| < 1$ in some absolute sense", it's that we have a useful definition of equality that formally obeys familiar properties of real arithmetic, and it's therefore convenient to speak of the series on the left as having a numerical value.
Is there some universal definition that separates all formal (real, say) series into "truly summable" and "truly divergent"? Perhaps, but it's easy to define a family of series whose convergence is algorithmically undecidable (e.g., generate terms of $1$ for each non-terminal step in an algorithm, and $0$s after the algorithm terminates, so that the series converges if and only if the algorithm terminates). It looks to me that the detailed workings of such a "universal convergence test" are outside human comprehension, and are certainly uncomputable in a practical sense.
