Why you can't abolish corporate tax
If companies always paid out their profits as dividends that would be fine. You could tax the dividends at whatever rate you feel should be taxed, and ignore taxing the corporations and save tons of hassle. The dividend rate would be important, because one of the reasons for double taxation at the corporation and personal level for corporate income is to pay for the liability shield the corporate structure affords its shareholders (versus a sole proprietorship or partnership where the owners have unlimited liability)
The main problem though is what if they don't ever pay dividends to allow their owners to avoid paying the taxes? The value of the stock would increase over time to reflect the greater assets (cash or whatever they used that cash to buy) and the owners could choose when to take their money by selling the stock, instead of getting regular dividends and regular tax bills. Especially in places where the long term capital gains rate is lower than that of normal income, that would basically guarantee the rich getting richer at even a faster rate (than they already do) than the poor and middle class do - who can't choose to delay taxes or get a special lower rate on their income.
Look at Berkshire Hathaway. It is a huge holding company (would probably be considered a hedge fund if Buffett had started it in 2002 instead of 1962) that consists of many dividend paying stocks like utilities and railroads. However, Berkshire Hathaway has never paid a dividend. It uses its excess cash to buy more companies, thus the stock is now $200K/share. In all those dystopian futures were one or a handful of corporations own everything, what would be the best way to get there? Don't tax them at all, so they keep buying up other companies until there are only a few, or one, remaining. Think of the lobbying power large corporations have today, even though there are thousands and thousands of multi-billion dollar corporations in the world. Imagine the power they'd have over governments if there were only five, or one. They'd own the governments, own the military.
The other problem, even if they did pay out their profit as dividends, is what about shareholders who live overseas? Do you make them pay US taxes on dividends from a US company? If you do, non residents might prefer to own shares in companies based in countries that have lower or no taxes on dividends, causing less investment in the US and a weaker dollar.