1. Media reports always get it wrong blah blah blah
2. The problem is much deeper than you indicate. Classical mechanics forces an ontology onto us that includes particle, wave, force, kinetic energy, potential energy. When people are taught/learn quantum mechanics they try to stick to that same ontology. But it doesn't work. Quantum mechanics requires a different ontology. Some things are shared by the two ontologies, like kinetic/potential energy. Others like force/wave/particle are not part of the quantum ontology. Nobody is upset that you won't find the word force in a quantum mechanics book. But people continue to use wave and particle, which they should not.
3. In quantum mechanics, the state of a system is simply a complex function over the configuration space of the system. Sometimes the configuration space is continuous (like position is Real^3), other times it is discrete (like the spin). This discreteness is why it is not correct to use wave to describe the state of the system. And also, just use a different name like "quantum state" for a different mathematical object altogether and not get confused.
4. This is not hard to implement in practice. I am teaching a course on quantum mechanics, and I told my students in the first class that we will never use the words wave and particle in the course (and why not), and we have stuck to that promise still. Similarly, we have covered why saying "an electron is both here and there" are statements that you don't make within the framework of (pure state) quantum mechanics. This is a good summary of the latter concept https://www.smbc-comics.com/comic/the-talk-3