Can I interpret logistic regression coefficients and their p-values even if model performance is bad? I trained a logistic regression model with two binary variables $x_1$ and $x_2$ where my target binary variable is $y$:
$$\log\left(\frac{p}{1-p}\right) =\beta_0+\beta_1x_1+\beta_2x_2$$
Looking at p-values, we notice that $x_1$ has statistical significance (p-value < 0.05).
However, the model performance is terrible (pseudo-R-squared = 0.004552).
So I would like to know if it is reasonable to conclude that there is evidence that $x_1$ is related to outcome $y$ or not.

Note: I am using the statsmodel Python library

I would tentatively say, "yes". Remember that pseudo-$R^2$ is an analogue to $R^2$, which in OLS is variance explained. Pseudo-$R^2$ is not that, but it's roughly similar. So your low pseudo-$R^2$ could mean you're missing an explanatory variable, a random effect etc. (many reasons possible). In other words, it's a not very good model for describing the data. It would thus likely be a poor predictive model. But, if you're just doing a hypothesis test, then even though you have a lot of extra variance that you can't explain, we're still finding a significant difference between the two x1 groups.
Again, I'm just saying this tentatively. If you want to know whether you should use this model, it also depends on your study aims and your analysis methods. Do you need an predictive model, or are you just looking to see if x1 has an effect? Do you need to control for multiple comparisons? etc.
