Coming back to my own question after I replicated the paper successfully for my thesis, where I found that my resulting SDF is always strictly positive and hovering around the value 1, just as expected given the formulation. Then, I also looked at their data and code and realized that this formulation is maybe just one way to "enforce" No-Arbitrage (NA). Because at the thesis presentation a professor asked me:
"How do you actually guarantee NA in your code?"
I don't, and they do not as well. Specifically, the law of one price (LOOP) implies the existence of at least one SDF that satisfies
$$ \mathbb{E}_{t}\left[M_{t+1} R_{t+1, i}^{e}\right]=0 \quad \Leftrightarrow \quad \mathbb{E}_{t}\left[R_{t+1, i}^{e}\right]=\underbrace{\left(-\frac{\operatorname{Cov}_{t}\left(R_{t+1, i}^{e}, M_{t+1}\right)}{\operatorname{Var}_{t}\left(M_{t+1}\right)}\right)}_{\beta_{t, i}} \cdot \underbrace{\frac{\operatorname{Var}_{t}\left(M_{t+1}\right)}{\mathbb{E}_{t}\left[M_{t+1}\right]}}_{\lambda_{t}} $$
whereas the absence of arbitrage opportunities (NA) in incomplete markets is equivalent to the existence of at least one strictly positive SDF. Hence, they aim to estimate one of possibly many strictly positive SDFs.
Given that they do not guarantee explicitly NA in their code, I assume that the choice
$$ M_{t+1}=1-\sum_{i=1}^{N} \omega_{t,i} R_{t+1, i}^{e}=1-\omega_{t}^{\top} R_{t+1}^{e} $$
was made such that the resulting SDF is very likely positive all the time and hence a suitable candidate SDF in incomplete markets with no arbitrage opportunities. Of course, since $\omega_{t}^{\top} R_{t+1}^{e}$ is a return, it may exceed 100% at some point in time, which would lead to a negative SDF. It is unlikely, but not impossible. Just my best guess at this moment.