I think "Tax the rich" is the wrong slogan. What we need is to incentivize sharing the progress we are making in terms of productivity widely instead of all the benefits going to a few. For example some people are talking about giving company shares to all workers. This would align the incentives for all workers in the company to do better and then benefiting instead of just the top few percent.

I wonder if this could be achieved by changing the tax code.

The moment tax policy goes into a wonky side alley it can be bought and paid for. Frontal assault is the only strategy that can possibly work for this one.