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 wanted to add a visual example reproducing results similar to the OP. (I usually understand things when I write the code for them).
Here we have 2000 datapoints. The true relationship between x and y is $y = 0.1x$, but the data is very noisy:
set.seed(1234)
N <- 2000
x <- seq(1, 5, length.out=N)
b <- 0.1
y <- b * x + rnorm(n=N, mean=0, sd=1)
plot(x, y, pch=19, cex=0.5)


A large amount of variation remains unexplained, $R^{2} \approx 0.02$. However, the linear model correctly gives a good estimate of the regression coefficient (~0.12) which is convincingly different from zero ($p \approx 10^{-9}$):
fit <- lm(y ~ x)
summary(fit)
Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.05891    0.06156  -0.957    0.339    
x            0.11762    0.01915   6.143 9.76e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.9893 on 1998 degrees of freedom
Multiple R-squared:  0.01854,   Adjusted R-squared:  0.01804 
F-statistic: 37.73 on 1 and 1998 DF,  p-value: 9.761e-10

So it's ok to interpret this model since the assumptions of the linear regression are satisfied. However, what you want to take away from it is a different question that depends on your context.
