People are living longer after retirement (over 10 years longer since the 1970s) and birth rates have fallen. Expenditure surpasses income eventually. There are other factors but that seems to be the main issue.

Making the pension a public problem won't fix it. The money has to come from somewhere.

The money can come from tax. You can balance it such that payments don't exceed tax income if balance is of concern.

No, it can't come from tax in general. The whole problem is a lack of productivity per capita. So any tax high enough to pay the shortfall will be oppressive. It's not a magic money machine that beats private investment earnings.

Your statement only makes sense if you assume growth. At steady state investments into e.g stocks and direct transfers is about the same thing.

And ye, if the economy shrinks so will pensions. But that would be the case for stock investments too.

If you invest wisely, you will make money or approximately tread water even if the economy is shrinking, because businesses operating with sufficient foresight make money on average. It might not be enough to retire on. That is a separate question. If there is not enough juice in the system to fund everyone's retirement, then neither investment nor taxation can create that productivity. There might be some in-between solutions but anything that does not involve generating enough production and profit to at least fund pensions seems guaranteed to be regrettable.

Re: Taxation, if you want to make up for the shortfall in people's pensions with an additional tax, you're assuming that someone else is making enough money to make up the difference. That is not the case. People can hardly fund their own pensions to the extent recommended, much less anybody else's. Most businesses are not profitable enough to fund extra pension contributions either, and imposing that cost will stifle them.