I'll admit that Biden doesn't speak as much about the wealth inequality and the need to tax the rich more as others. But he does bring it up now and then
I don't really understand your logic on this.... Let's use Bernie Sanders as an example. He is wealthy now (mostly off book deals)... but is a champion for socialist causes and indeed labels himself a democratic socialist.
The first thing you have to address in this situation is the notion that because you want a different tax system, wealthy people to pay a higher tax, end loopholes, etc that you shouldn't have any money yourself. You are basically saying "Bernie is rich, but wants to enact policies that take money from the rich, therefore he is a hypocrite".
Bernie has tons of problems with capitalism, but he has made a lot of money. But did he make a lot of money because of capitalism?
Lots of people in the countries Bernie wants to model after have rich people.... Jeff Bezos would still be extraordinarily wealthy, even if we taxes 90% of his earnings after a certain amount. You would still have lots and lots of rich people. Now, what some want to do is basically tax billionaires out of existence, and simply make them hundred millionaires. That's a tax argument though, not a capitalist argument.
Example, by your logic, the US is basically the only "capitalist" country on earth... Every other developed nation is a socialist one, because they have things like wealth re-distribution (via tax policy), universal healthcare, education benefits, etc. Even bastions of free market capitalism like Singapore and Hong Kong have these things.
All of these countries that Bernie wants to model after, mostly like Sweden, Norway, and Denmark... have market economies. That is what makes you capitalist. The businesses operate in a free market, and are owned by private citizens.
What truly makes you socialist is when the state runs the business. That is like China. Although China is doing more of a hybrid system these days, where the govt has influence over "private" companies and has great power over them.
So, Norway has a market economy, private business ownership, etc. They are a capitalist country. Now, what they have is a robust democracy, and they have a different tax system and social welfare system than we have. Doesn't mean they are a socialist country, it just means they do a lot more of the things we also do. We have universal healthcare, but just for old people. We have UBI, but just for retired people... We have public schools... We have a public postal service... Some of our citizens get to go to school for free, etc. They just do a lot more of these things, but they do so within a market economy.
Conservatives in this country either believe or have been led to believe that any/all of these things makes you a communist. That isn't true at all. Again, by that definition, every other country on the planet is a communist state. We know that isn't true. All these places have open and free market economies, they just do a lot more with their tax revenue. They spend a lot less on military, and a lot more in the community. They do have aspects of socialism in their country, because they do have some state-run industries. Those aren't inherently bad. Where you can go wrong is when ALL or too much industry is run by the state. Then you start getting into real communism/socialism. But the things we are talking about don't even being to approach that level, in my opinion.
Just my thoughts....