Even the most socialist countries usually have the idea of “my bed”, “my house”, etc. There’s still some form of ownership, and market driven prices>
You are mistaking "private property" and "personal property". Socialism has nothing to do with personal property, such as your personal posessions that you actually use like your house, toothbrush, car whatever. Socialism wants to abolish private property, aka private ownership of the means of productions, like factories, schools, public transport and etc. As a CEO of a bus company might not necessarily even use it, it is used by the commons therefore it should be owned by the commons; your toothbrush is not used by the commons.
I don't think any socialist is making a big deal about socialism just "because it's socialism", nor any ideology to be honest. People first look at what that ideology entails, and then make judgements about it. A lot of people nowadays are seeing the contradictions of capitalism and see no future under it and crave some kind of alternative, however no current neoliberal, nor social democrat movement is giving an answer to the contradictions people see. Socialism however directly answers and explains the contradictions through marxist class dynamics.
And I dont get your point about labels. Labels are useful abstractions to summarise schools of thought. It is easier to say "I am a socialist" than to say "I believe in....:" and then just repeat the entire volume of das kapital.
I suppose yeah I understand what you mean. Especially in the US there is a lot of political tribalism and extreme polarisation, and honestly it has one of the most extreme cases of anticommunist propaganda where something as basic as free healthcare will be seen as "evil communism". Regarding your article, go for it. Clearly you identified some kind of issue that needs addressing and that article you want to make is your reaction to it. What I am trying to say is that the fact that you even want to make the article is already its justification :)
Also slightly unrelated but this reminds me of "radical centrism" where you basically take any extreme idea that sounds good no matter what side its from, be it communism or nazism.