The statistical interpretation of the coefficients doesn't depend on how the model was fit. I could make completely random guesses of the coefficients and they would have the same interpretation as they would had I estimated them with maximum likelihood. For two units identical on all measured variables except that they differed on $X_1$ by one unit, the difference in the log odds of success is $\beta_1$. That interpretation comes directly from simply writing down the regression equation and has nothing to do with the fitting process.
To interpret the coefficients as consistent estimates of some "true" association, or as total effects rather than direct effects, or as causal effects rather than mere conditional assocations, requires more assumptions, far more than whether the model fit well in your sample.
For example, let's say the true data-generating (i.e., structural causal) model was
$$P(Y=1|X_1,X_2) = expit(\gamma_0 + \gamma_1 X_1 + \gamma_2 X_2)$$
Let's say I'm considering the model
$$P(Y=1|X_1) = expit(\beta_0 + \beta_1 X_1)$$
which excludes $X_2$. $\beta_1$ doesn't have a causal interpretation, but it's the regression slope you would get if you were to fit that model to the population data (i.e., so there is no sampling error). The interpretation of $\beta_1$ in this model is: For two units that differed on $X_1$ by one unit, the difference in the log odds of success is $\beta_1$.
Let's say I collect a sample and then pull an estimate of $\beta_1$ out of a hat and call it $\hat \beta_1^{guess}$. Even though that value is completely unconnected to the sample, it still has the same interpretation as any other estimate of $\beta_1$, which is as an estimate of the difference in the log odds of success for two units that differed on $X_1$ by one unit. It's not a valid or consistent estimate, but it's an estimate of a quantity that has a clear interpretation. The quantity ($\beta_1$) does not have a causal interpretation, but it's still meaningfully interpretable as an associational quantity.
If I were to estimate $\beta_1$ with maximum likelihood, and call the estimate $\hat \beta_1^{MLE}$, it has the same interpretation as $\hat \beta_1^{guess}$, which is that it is an estimate of $\beta_1$, which, again, has a clear interpretation. $\hat \beta_1^{MLE}$ is a consistent estimate of $\beta_1$, so if I were to want to know what $\beta_1$ was I would be inclined to say it's closer to $\hat \beta_1^{MLE}$ than it is to $\hat \beta_1^{guess}$. $\hat \beta_1^{MLE}$ could result from a terribly fitting model, and that would say nothing of its interpretation. A terribly fitting model might result because we failed to include $X_2$ in it. That doesn't change how $\beta_1$, and thus how $\hat \beta_1^{MLE}$ and $\hat \beta_1^{guess}$, are interpreted.
If you wanted to interpret a regression coefficient as causal, then you want to estimate $\gamma_1$, not $\beta_1$. The interpretation of $\gamma_1$ is the change in the log odds of success caused by intervening on $X_1$ by one unit while holding $X_2$ constant. Any estimate of $\gamma_1$, regardless of how it came to be, could be interpreted as an estimate of the change in the log odds of success caused by intervening on $X_1$ by one unit while holding $X_2$ constant. You could even use $\hat \beta_1^{guess}$ as an estimate of $\gamma_1$ and it would still have this interpretation. It would likely be a bad estimate that you shouldn't trust, but that doesn't change its interpretation. Even if you estimated $\gamma_1$ using maximum likelihood estimation of a model that included both $X_1$ and $X_2$, its interpretation would be the same; it would likely just be a better estimate (but it doesn't mean it's a good estimate!).
All this is to say that the interpretation of coefficients comes from the model as it is written, not the way they are estimated or how well the estimated model fits. These may serve as indicators as to whether the estimated coefficients might be close to the population versions they are trying to approximate, but not how they should be interpreted. For example, a poorly fitting model resulting from regressing $Y$ on $X_1$ may indicate that $\hat \beta_1$ is a poor estimate of $\gamma_1$, but it may be a good estimate of $\beta_1$. The interpretations of $\beta_1$ and $\gamma_1$ are unrelated to how the estimates were generated, and the interpretation of the estimates is simply as estimates of those quantities.