The argument that amassing wealth is bad is fundamentally untrue, as it depends on the skill of the person amassing the wealth (I gave an example of one type of billionaire that probably wasn't bad, it's also an example of one type of wealth amassing that isn't bad). Concentrating power is good if the concentrating is happening in people that will push the production possibilities frontier out quickly. Amassing wealth through market power abuse is bad (looking at you Bezos and your company amazon since atleast 2005, with various nefarious business practices that are obvious market power abuse rather than innovation as the example of this). So so wealth amassing is good, some is bad, the feature of bad wealth amassing is market power abuse, so my criticism is we shouldn't hate on the wealth concentration, we should hate on the method of wealth concentration that cause deadweight loss and we have ample laws to do that via antitrust, we just aren't using them effectively since roughly the 1970s (thanks Rhenquist supreme court).

I never said money is always better spent by billionaires, I made the distinction I reiterated and expanded on above and said government has a lifecycle so where in the lifecycle a government is determines how likely it is to spend money better than a billionaire (or better than any individual for that matter), but again that comparison is individual because there are differences among individuals and governments and the devil is in the details. I have the most experience with the USA and Canada and I would say both of those are corrupt and wasteful enough to give serious pause to any desire to send them any additional money but they are both still productive enough that the inefficiency and waste isn't necessarily obvious to everyone yet (it's probably coming, wait a decade or two and if you don't see a mass downsizing of the federal labor force or other large reduction in spending with a smaller reduction in services to stave off the obviousness it will likely be obvious to all by then.)

Certainly they do those things, how they do those things determines whether the government is efficient or wasteful. Whether they are beneficial or highly profitable is up for debate because none of the words you mentioned are actual things, they are amalgamations of good and bad things and the ratio matters a lot. I'm a fan of progressive income taxes as I like paying more if I get more benefit and I certainly benefit from effectively provided healthcare, roads, education because my workforce gets more productive and my customers get richer and more able to buy stuff. This isn't the issue here, the issue here is understanding how likely more money to government will benefit society or benefit politicians and harm society and understanding that most of the problem with billionaires isn't a billionaire problem, it's an antitrust problem, for which we have a legislated solution that we need to grow up and start using again (break up the large company exercising market power causing deadweight loss, regulate business practice to outlaw stupid anticompetitive crap like most of what makes up a seller contract with amazon these days).

The concern is also that piketty fundamentally misunderstands the effect of a wealth tax on capital markets. they will be forced to sell stock/companies to pay it and that will crush the financial market. Of the group of billionaires that are actually doing a good job pushing the production possibilities frontier out, it is also actively harmful because it diminishes their control of their companies and harms company financing capability. The system shock of somthing like a 10 or 20% wealth tax will cause a lot of deadweight loss that Piketty does not appreciate while doing nothing more than funneling money into questionable governments. If one is going to disrupt the financial market like that one should at least be doing it mitigating market power by, for example, breaking up the obvious oligopolies that infest our business environment.