This is weird, could you perhaps send us the problematic file to contact@esotericsoftware.com to test the vertices? We don't need the images if they're under NDA, but just an indication of the problematic mesh to understand if the cause is a bug.
Thank you!

Thank you for sending the file,

I could replicate the behavior and I've opened an issue to further investigate it here:
Vertex could not be deleted · #480 · EsotericSoftware/spine-editor

We will update you when we find the reason, in the meanwhile, it seems like moving or deleting one of the other edge vertices makes deleting that vertex possible.