Binary outcome in randomized controlled trials -- OLS or logistic? I'm running a randomized controlled trial which has good balance in the co-variates. I'm unsure whether to use:


*

*OLS: $P(Y_i=1) = \beta_0 + \beta_1 \text{Treat} + \epsilon_i$.


This is problematic because the variance of binary $Y$ is not homoskedastic, thus invalidating the OLS assumption. Note that the right-hand side (RHS) only has the 0-1 indicator $\text{Treat}$, so I don't have the problem of the RHS going beyond the 0-1 range.
$\beta_1$ in this case is simply the difference in sample mean between the control and the treatment group.


*

*Logistic: $\text{logit}(P(Y_i=1)) = \beta_0 + \beta_1 \text{Treat}$


This is problematic because the correct model needs to include other co-variates (despite balance) and not just the $\text{Treat}$ indicator. The substantive effect of $\beta_1$ on $P$ then depends on the value of these other co-variates. However, then I no longer see the benefit of my experimental design, which allows me to not control for any other co-variates in the OLS case.
SUMMARY of the answers I've got so far:


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*@glen_b suggests that I should include predictors in both OLS and logistic despite experimental design in order to improve precision. In this case of binary outcome, including other predictors means that OLS is not an option anymore (because the RHS now can go outside the 0-1 range)

*@AMD suggests how to interpret the treatment effect in logistic regression, regardless of whether other predictors are included:
$$
\frac{p}{1-p} = \exp(\beta_0 + \beta_1 \text{Treat} + \beta_2 X)
$$
When $\text{Treat} = 0: \frac{p}{1-p} = \exp(\beta_0 + \beta_2 X)$
When $\text{Treat} = 1: \frac{p}{1-p} = \exp(\beta_0 + \beta_2 X) \times \exp(\beta_1)$
Therefore, the treatment effect is that it multiplies the odd-ratio ($\frac{p}{1-p}$) by $\exp(\beta_1)$. Pro: the size of this effect does not depend on other $X$. Con: This is not directly about $p$, which some may be interested in.
REMAINING QUESTION due to several conflicting advices: If I don't care so much about precision and just want to prove the causal effect of my treatment, is it okay to not include other predictors in the logistic given the experimental design? If I fail to include some interactions (with observable and/or unobservable), is $\beta_1$ of $\text{Treat}$ still a consistent estimate of the causal effect?
 A: Edit for clarity: It looks like my responses here have led to some clarifying additions to the question or additional information in comments,  which make parts of my answer now at least partially obsolete. However, I plan to leave my answer as is, partly for context and partly because I believe the points raised may be relevant to later readers.
Changing the order a little:

Logistic: ... This is problematic because the correct model needs to include other co-variates (despite balance) and not just the Treat indicator. 

Both models should include predictors that would be likely to have a substantive  effect, even if the design is perfectly balanced and there are no interactions between variables. For example, to omit them if you have them would reduce power - for example, in OLS it inflates error variance by incorporating their effect in the error. 
[Further, if there can be interactions between variables, you won't get the expectation in the model right. You should be consider diagnostic checks for potential interactions with such variables, included or not.]

OLS: ... This is problematic because the variance of binary Y is not homoskedastic, 

That's not even the worst problem with OLS on this. The even more serious problem is that once you include the other covariates*, the relationship cannot be linear -- you will - necessarily have a model that predicts probabilities that are negative and others that are greater than 1 (predicted rather than fitted).
*(which I strongly believe you should, unless you are confident they are actually unrelated to $Y$)
A: A lot of economists use linear probability models, arguing that LPM provides the linear approximation of the conditional expectation function, which is often considered "good enough."  Consistent (in large samples) standard errors can be gotten by using ``robust'' variance-covariance matrix estimators.  
This is an OK argument if you really just want $\beta$, and want it to be interpretable as a conditional expectation in the larger group.  You don't want to do this if you have any interest in prediction.
In reality though, arguing that $\beta$ increases a probability by a certain amount can only make sense on average (hence conditional expectation in the sample, which you generalize to the population).  It can't be a description of what you would expect to happen to unit $i$ if you treat them.  Because if $i$ has covariates that push them up or down, then adding $\beta$ to the effect of those covariates could lead to probabilities outside of 0/1, which wouldn't make any sense.
That said, logit models involve assuming that the link between the predictors and the outcome is a logit.  This can be restrictive.  
But you can interpret a simple logit coefficient as an odds ratio by exponentiating it.  For example, if $\hat\beta=1$, then you're estimating that the treatment leads to a $e^1=2.7$-times more likely odds of $y$ equaling 1.
