Coming back to my own question after I replicated the paper 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

$$
\mathbb{E}_{t}\left[M_{t+1} R_{t+1, i}^{e}\right]=0 
$$ 

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 

$$ M_{t+1}=1-\sum_{i=1}^{N} \omega_{t,i} R_{t+1, i}^{e}$$

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.