Eliminating the income cap will not generate enough revenue to cover the shortage. It isn't even close.

A privatized plan could still provide a minimum guaranteed income to all workers. (the "workers" part there is important, IMO).

If you take the FICA taxes that I generated (taken directly from my paycheck and my employer's "share"), and pretend that it was invested in a variety of index funds with a long-term growth of 5% (actual long-term growth is closer to 7%), I should have a bit over a million dollars at retirement age (+/-65). That includes the various "crashes" that have happened over the years.

If my retirement income is kept to less than long-term growth, I would have roughly $50k per year. Right now I'm projected to get $20-$25k depending on when the "haircut" hits and I have to work until almost 68.

Make it so half of any unused funds go back to the feds to cover those with lower lifetime earnings.

Even a person that works for nearly the minimum wage can generate $500-700k of retirement. But we can sweeten that a little from the government.

In the case of people with significant old-age issues (i.e. dementia) there is a pile of money that can be used for their care. That reduces the amount of money that the government must pay.

Social security as currently established is a Ponzi scheme. Sweden has privatized successfully as has Columbia. The only thing Social Security does well is to ensure that seniors are as equally poor as possible.