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

It is routine in many areas to do inference with models that have low performance. I cannot think of any examples with $R^2$ values as low as the pseudo $R^2$ you have, but I know I've read papers in high-end journals that do inference on models with linear regression $R^2$ values around $4\%$, maybe even lower.
A cause for concern is if you've missed something in the modeling. For instance, it could be that there is a generally upward trend, but only because the data form a checkmark shape. In that case, your model is misleading. For low values of your feature, increase the value actually decreases the outcome, and you might find yourself frustrated if you apply your bogus model to such values only to find the outcome decreasing when the regression says it should increase.
However, a good regression modeling strategy should help you avoid such a situation, so if you have been thorough in your modeling, the low performance need not be interpreted as the regression screaming, "I'm worthless," at you.
