Rentier capitalism vs investment capitalism? A company has backers who put in money seeking a return from either share-price growth through company growth, and yearly rent in the form of dividend payments. Debt is clearly the case where share price growth is 0 and only the yearly rent is paid, (Socialism is 1/population for both + a wage). Why do stockholders get both the right to make money from the investment, and the right to extract rent at the same time? How does having both rights at the same time impact the economy? Is there any economic thinker that considers banning dividend payments on shares from a legal perspective?
A share is an entitlement to the ownership of a company. Such ownership brings several benefits like voting rights and dividends, and the owner is free to sell such ownership whenever s/he wants, making a capital gain (or a loss) in the process. 

Why is this the case? Well, it was like that with the first publicly listed stock-owned company (the Dutch East India Company). In terms of incentives, it fosters the allocation of (financial) capital to new ventures by giving those taking the risk to benefit from it. It has proven a highly successful legal arrangement to foster businesses and investment. 

Another common legal arrangement is that of non-profit corporation. These organisations are usually started by a group of people, which contribute capital to it (albeit this is not represented in the form of shares). Sometimes they transfer the ownership to a foundation. In any case, these companies cannot distribute profits to the owners.

Finally, as you say an alternative legal arrangement would be one were ownership of shares do not lead to income via dividends, but such shares still allow for capital gains. In fact, some shares pay no dividend at all (as a matter of fact, the taxonomy of shares is quite diverse). As this article explains:


  investors who buy stocks that do not pay dividends prefer to see these companies reinvest their earnings to fund expansion and other projects which they hope will yield greater returns via rising stock price.


So in principle you could set up a company that does not issue dividends in order to foster its growth. Naturally, this could be changed in the future through a motion in an annual general meeting of shareholders. There are also plenty of companies who decide not to pay dividends in order to foster expansion (e.g. Apple until a few years ago). Yet, many of these companies, and particularly in the last decades, have decided to execute share buybacks rather than foster expansion and investment. This is, the company buys their own shares with its profits, which fosters price growth and hence expands capital gains. This is part of a wider short-termist trend that some have denominated "Quarterly Capitalism". 

There is very interesting new research about the consequences of this "quarterly capitalism" on investment and growth. Check here, here, and here. The latter concludes:


  We show that capital no longer flows more to the industries with the best growth opportunities because, since the middle of the 1990s, firms in high q industries increasingly repurchase shares rather than raise more funding from the capital markets. 


This evidence gives a rather counterintuitive result (at least from the normative meaning of the words), which is that a rentier capitalism might be better for investment and growth than an investment captialism.