In a binary-dependent variable model (logic) will the inclusion of a scale introduce endogeneity? This question is somewhat related to my previous question, see here: https://stats.stackexchange.com/questions/463221/why-when-do-attitudinal-scales-introduce-endogeneity

I wonder why a scale as an explanatory variable will always make a binary-dependent model endogenous?

This is what an editor told me along with rejecting my paper, and as long as I don't change it/acknowledge the possible endogeneity I cannot resubmit the paper.
Based on the clarification in your comment this is not because you use scale variables but because the scale variable that you use is self-reported.

It is well known in literature that self-reporting leads to endogeneity. See for example: Lindeboom, M., &amp; Kerkhofs, M. (2004). Subjective health measures, reporting errors and endogeneity in the relationship between health and work.

The reason why that is when people self-report some outcomes those are often affected by other unobservables that you do not include in your model which is a form of endogeneity. For example, studies that measure happiness show that the self reported happiness can vary depending on which time of the day/week/season you make the measurement. 

Moreover, you can have also endogeneity in more narrow sense unemployed people can self-report that they cannot work as a rationalization of their unemployment status and hence you would get reversed causality where the unemployment actually causes the self-reported inability to work even though the person actually would be able to work as measured by objective measurable criteria (in a model where employment is regressed on self-reported ability). 

So it’s not that there is any problem with having scale variables. For example, having some 1-10 index but rather the problem is that in your case the scale was made from self reported data.