Why are my coefficients too large when control variables are not added?

Your analysis is not suitable for the "classical" difference-in-differences (DD) approach. It is, however, amenable to the "generalized" DD framework.

Here is your model:

$$ text{ln}(y_{pt}) = gamma_{p} + lambda_{t} + delta T_{pt} + theta X_{pt} + epsilon_{pt}, $$

where you observe programs $p$ across days $t$. $gamma_{p}$ denotes "program" fixed effects. $lambda_{t}$ denotes "day" fixed effects. Your dataset would include a series of $P-1$ indicators for programs and a series of $T-1$ indicators for days. This might be a lot of dummies to append to your data frame, so I recommend letting software do most of the heavy lifting for you in terms of estimating your program and day effects.

The policy dummy $T_{pt}$ should equal 1 in all 'program-day' combinations where the policy/treatment is in effect, 0 otherwise. This, in essence, is your interaction term (i.e., $T_{pt} = text{Treated}_{p} times text{After}_{t}$). I do not recommend interacting the constituent terms in this equation. Rather, create the interaction variable manually. In other words, if a program $p$ is treated program and it is in the "after" period then set it equal to 1, 0 otherwise. For programs never treated, they would be equal to 0 in all days. Note, this is the equivalent to the policy variable in your toy example. It is also equivalent to the Treated*After column as indicated in your data frame. Again, simply regress your outcome on a full set of program dummies, a full set of day dummies, and the policy variable.

I actually think you understand the "generalized" DD framework well. I think you're caught up on why the DD coefficient in this setting does not mirror the DD coefficient in the "classical" setting. In the "classical" DD framework, you must have a well-defined before and after period. I am surprised software returned a DD estimate for you in your second run. Think about why this wouldn't work. In the "classical" case, your $text{After}_{t}$ variable should be coded 1 in the post-treatment period in both treatment and control groups. It would represent the simple passage of time for the control group in the absence of treatment. But you don't have that here! In fact, your $text{After}_{t}$ period indexes days when treatment 'switches on' for treated programs. Thus, when $text{Treated}_{p} = 0$ and $text{After}_{t} = 1$, what is the before and after change? If I am correct, the values would be 0 for all control programs (i.e., never treated programs) in the days before and after treatment. Or, suppose they aren’t all 0’s. What is the “after” period for control programs? If I am making any sense, then it should be obvious that the “after” period is not standardized; treated programs ‘switch on’ early or ‘switch on’ late. In sum, $text{After}_{t}$ should not be used in the latter (base) model. The "after" period is clearly not well-defined in your setting; it varies across programs. You must proceed with a "generalized" DD approach.

Remember, you created the $text{After}_{t}$ variable so you could interact it with $text{Treated}_{p}$ and obtain your policy variable (i.e., $T_{pt}$). But that $text{After}_{t}$ period is not the post-period we use in a "classical" DD analysis! In staggered adoption settings, eschew the standard approach and proceed with the "generalized" DD equation I specified above. It is more versatile and will handle irregular treatment exposure periods.