So you should ask yourself what the p-value on the intercept tells you. Does a p-value greater than .05/.1/.2 tell you that the estimate is zero? Or does it tell you that you're not sure if the estimate is positive or negative. The second answer is the right one.

That said, you should never use the p-values on your coefficient estimates to determine which model you are going to use. Instead, you should use model selection criterion to determine which model is the best at (presumably) forecasting new data. In this case, determine if the model with the intercept is better/worse than the model without the intercept. I would use out-of-sample metrics like cross-validation, but I've seen BIC/AIC perform equally well. I would not use any type of r-squared or strictly the likelihood to choose the model.

That said I would say that you should almost certainly leave the intercept in. What you are saying by leaving it out, is that the probability that dependent variable equals one when all other variables equal zero is exactly 1/2. This probably doesn't make sense and is basically impossible.