"Bad" control variables in randomised treatment trial I am analysing the effect of a randomised treatment on several outcome variables.
First i am interested in whether the treatment changes the first outcome (non-pecuniary value) by controlling for demographics characteristics X.
$$ Value = \beta_0 + \beta_1*Treatment_i + \beta_2*X_i + \epsilon_i $$
However, I am also interested if it affects another outcome (pecuniary value) that is according to the literature connected to the first variable $Value$. Therefor, I originally intended to control for it to avoid omitted variable bias.
$$ Money_i = \alpha_0 + \alpha_1*Treatment_i + \alpha_2*Value_i + \alpha_3*X_i + \epsilon_i$$
Now lastly, I am interested in the effect of the treatment on a third outcome variable (Probability to go to school) that is according to the literature related to the variables $Money_i$ and $Value_i$. Hence, therefor I thought to control for both of them in the regressions.
$$ Probschooling_i = \gamma_0 + \gamma_1*Treatment_i + \gamma_2*Value_i+\gamma_3*Money_i + \gamma_4*X_i + \epsilon_i$$
However, now I came across the concept of "bad" control variables that bias inference. In my case, as the variables $Value_i$ and $Money_i$ are themselves dependent variables in the other regressions, I thought that therefore they can be called bad control variables (some kind of dependencies). My question now is if that is true? Are they bad control variables so that i do need to take them out (and to not bias the treatment effect) ? Or can I just leave it as I have it (and so to avoid omitted variable bias)?

The answer would probably depend on what causal structure you are assuming. To take your example with the Money variable: if you assume that treatment affects both Value and Money, and Value also affects Money, then whether to control for Value or not depends on your question!
e.g.:
dag <- dagitty::dagitty("dag {
                 Treatment -> Value -> Money
                 Treatment -> Money
                 }")

plot(dag)

If you are interested in the total effect of Treatment on Money -- i.e., does Treatment increase Money in general -- then controlling for Value will invalidate your inference, because you will be "cancelling" one important way in which Treatment does cause and increase in Money, namely because it increases Value, which then itself increases Money.
If you are interested in the direct effect of Treatment on Money -- i.e., does Treatment increase Money "directly", without counting the path that goes through Value? -- then you should control for value.
I find McElreath's explanations quite enlightening, in this lecture and the next: https://www.youtube.com/watch?v=e0tO64mtYMU&list=PLDcUM9US4XdNM4Edgs7weiyIguLSToZRI&index=5
Also there is a paper, geared more towards psychologists, that would likely help you wrap your head around this: https://journals.sagepub.com/doi/10.1177/2515245917745629
