In these lectures on the CP Problem or this SE post you can see how you can relate the value of of the theta angle $\theta$ to the value of the neutron eDM. A priori it should be possible to obtain a similar result for the proton but, as far as I know, we mostly focus on the neutron eDM because it is easier to measure. So, even if there were no $d_n$, you could still have other similar magnitudes that are not null.
But OK, let's imagine that there is some mechanism that suppresses all QCD eDMs (and other CP-violating quantities) without that requiring a non-null $\bar{\theta}$. Could we still have a CP problem? Well, it could still be that QCD does not violate CP-symmetry, but we would not be able to measure it experimentally, so we would lose all experimental justification to search for reasons that drive $\bar{\theta}$ to be small! This doesn't mean that axions, or whatever other solution to the Strong CP problem, don't exist. We would have just lost the biggest reason to believe they exist (there are others, as we think axions might be cold dark matter, see this again).