I think you're mixing terms. Central planning is the soviet model, it's not a feature of socialism. Socialism just means collective ownership of the means of production.
You also seem to be treating democracy like it's a binary. A country is either democratic or it isn't. But I gave two ways above in which capitalism and democracy can coexist, it's just that a decision needs to be made about which should be preeminent.
Based on your binary notion of democracy, you claim that long running democracies are all capitalistic. Setting aside the fact that "long running" means no more than two hundred and fifty years, I would argue that not all of these are as democratic as they claim.
We, right now, have Trump trying to compromise elections in the United States. Let's say that he succeeds and then goes on to a third term as supreme leader. No doubt there would still be elections, and no doubt there would still be claims that the country is democratic. I don't think that those claims would mean very much.
Even without Trump, there's a lot of uncertainty of how democratic the country really is. Before Trump there was already a 94% correlation between campaign spending and election success in the house. 82% in the senate. Almost 50% of the house are millionaires. More than 70% of senators. That should at least give you pause.
I thought I was pretty clear above that "unfettered" meant the limitless accumulation of wealth. And thus the concentration of power. Which is the thing that is antithetical to democracy. And it's something that we see in the United States, and something which has been growing slowly over time.
You seem to lack examples of alternatives. There's a Spanish organization called the Mondragon Corporation which operates on a
democratic-socialist model. There are American companies which are employee-owned, like Bob's Red Mill, but like capitalist companies they're organized on a central planning model and I don't know to what extent "employee owned" really reflects influence or if it's just a revenue share. The Mondragon Corporation is a socialist company. If Spain, as a country, wanted to go fully socialist then they could just force other Spanish companies to adopt the Mondragon model. That's all that it would take. This would not imply that Spain would also be a democracy. Even though Spanish companies would be individually democratic in this scenario, that does not necessarily mean the same thing about the country in which they reside. However, since the nation's companies do represent the economy in a practical sense, forcing them to go socialist is all that would be required to make the country socialist.