Is there any advantage to incorporating some sort of business for the purpose of holding and trading equity holdings?

An issuer company, for the purpose of investing, would be subject to investment company regulation. The advantage to this type of company is to raise funding from outside investors.

A non-issuer company, for the purpose of investing, would be subject to broker/dealer registration. The advantage to this type of company is to bring particular talented traders into one organization by rewarding the traders with percentage partnerships.

Now there is a 21% corporate tax rate but distributions from a corporate tax structure involve double taxation of distributions. Most trading companies use a higher-rate pass-though-taxation but avoid double taxation of distributions.

But an individual investor probably doesn't need a corporate structure because futures traders have an advantaged capital-gain tax rate while one-year equity holders also have an advantaged capital-gain tax rate.

Now an issuer company with an investment portfolio, but with a purpose other than investing, can avoid being classified as an investment company by having a portfolio that is 60% Treasury securities. Then if there is trading of the issuer shares, a 21% corporate tax rate might be chosen along with a policy of no distributions. Similarly, a non-issuer company could avoid being classified as a broker/dealer by having a purpose other than investing and by having a portfolio that is 60% Treasury securities.