Background
Let $\zeta(\cdot)$ be the Riemann zeta function. I'm looking for a visual proof of the infinite series identity $$\sum_{n=2}^{\infty} \left( \zeta(n)-1 \right) = 1. \tag{1}\label{1}$$
This identity arises in the study of rational zeta series.
While I've found a number of visual proofs of infinite series, some of which are described over here, and here, and another one in this question, I haven't seen or obtained any for this identity.
A Problem in Discrete Geometry
I believe this question is interesting by its own accord. In addition, however, I hope any answers to it might yield insights into a specific problem in discrete geometry. It concerns the question of whether a square can be covered by rectangles of size $\frac{1}{k} \times \frac{1}{k+1} $. One can read more about this problem on the following MO thread.
The problem hasn't been solved yet. Approximate solutions have been generated, however. Below is an example of such an approximation by MO user Ed Wynn:
The connection with this problem arises upon proving equation \eqref{1}. We have:
\begin{align} \sum_{n=2}^{\infty} \left( \zeta(n)-1 \right) &= \sum_{n=2}^{\infty} \sum_{k=2}^{\infty} k^{-n} \tag{2} \label{2} \\ &= \sum_{k=2}^{\infty} \sum_{n=2}^{\infty} k^{-n} \tag{3} \label{3} \\ &= \sum_{k=2}^{\infty} \frac{1}{k(k-1)} \tag{4} \label{4} \\ &= \sum_{k=1}^{\infty} \left( \frac{1}{k} \times \frac{1}{k+1} \right) \tag{5} \label{5} \\ &= \sum_{k=1}^{\infty} \left( \frac{1}{k} - \frac{1}{k+1} \right) \tag{6} \label{6} \\ &= 1 \tag{7} \label{7} \end{align}
We see that the connection arises in equation \eqref{5}, as the same terms as the sizes of the rectangles are indicated.
Please note that I am not looking for a visual proof of equation \eqref{6} -- \eqref{7}. This question is already addressed in the first one of the three MSE links provided above, which all consist of visual proofs.
Question: is there a visual proof of \eqref{1} ? It can be a proof of your own making, or a pointer to a relevant paper.
