No change in the coefficients of my time-specific variables of interest when controlling for demographic effects

Ciao Guys,
I have a balanced panel set and I study the following model:
lm(value ~ taskXjan20 + taskXfeb20 + taskXmar20 + taskXapr20 + demographic_criteria),

where value is a binary variable, being 1 if individual i has been unemployed in the previous month and 0 otherwise. It expresses therefore the probability of getting unemployed. I would like to explain this probability with the task group an individual is assigned to (which is a binary variable as well: 1 for the specific task group). This variable for a specific task group is interacted with the specific month (obviously being a binary variable as well). However, as I want to add demographic criterias as control variables (in total quite a lot ranging from AGE to Education,...) I do not see any difference for my coefficients.
Further, as I try to run a regression like that:

lm(value ~ task + taskXface_to_face + taskXremote + taskXessential + demographic_criteria),

where the interactors of my task variable are now time-independent variables, I see severe changes as I control them for my demographic effects.
My question now is, did I forget to adjust my regression for any further trends if my variables of interest directly interact with time-specific variables? And if yes, why did I not have to include them into the 2nd regression?

A model, which is similar to the first one, have been studied on page 11 and 37 of the following link.
Surprisingly he was able to generate different results when adding demographic control variables.

Hope somebody can help me!
Many thanks in advance, Freddy