Does a sound LPoE establish that god is impossible?

I've heard that claim, but do you think it is true? I don't. All LPoEs (Logical Problem of Evil's) can be seen as an inconsistent set of propositions. Here's a really simple version:

Simple LPoE: 1. God is all-good. 2. God is all-powerful. 3. God is all-knowledgeable. 4. If god is all-good, all-powerful and all-knowledgeable, then there is no evil. 5. There is evil.

The above set of propositions is inconsistent, i.e. they cannot all be true; it is impossible that they are all true. But from the fact that a set of propositions cannot be true, it does not follow that any one of them are impossible.

It does not follow either, that if all but one of them are true, then the last is necessarily false; impossibly true. That would be to commit a modal scope fallacy. What does follow from all but one of them being true is that the last one is false. So, there is a confusion between:

1. If all but one of the propositions in an inconsistent set are true, then the last proposition is necessarily false.

2. Necessarily, if all but one of the propositions in an inconsistent set are true, then the last proposition is false.

So, given the above I don't know why someone thinks that a sound LPoE establishes that god is impossible. For that to work, one would need to establish that evil is necessarily and I don't think that is feasible. After all, if evil is necessarily, it is not god's fault that there is evil, is it?