I haven't been able to find any information on this, so I hope you can help. It seems to me that corporate taxes are the worst kind of regressive tax and that corporations do not actually pay any tax, per se - we pay it for them. Corporations get all monies for taxes from the goods they sell. So the people buying those goods actually pay the tax, indirectly. The most obvious example I can think of is a gas tax. For every penny the state raises the gas tax, the price of gas goes up by that amount. It just happened recently here in New Jersey. People are driving less, so less tax revenue. NJ raises the gas tax to compensate and gas prices went up accordingly. And the people who can least afford it are forced to pay more for gas. Doesn't impact the millionaires - they won't even notice. Doesn't impact the corporation - their profit remains the same. Corporate taxes in this country (USA) remind me of the VAT in Europe. Same thing, but at least they are honest about it. We seem to like to live in this delusion that corporations are creating money out of thin air (only the Fed can do that) and so they need to be taxed more. It's amazingly popular, especially at election time. On the other side, if a business can not raise their prices because of market pressure, they end up going out of business. So what am I missing?
You have described an economic concept called tax incidence. There is a wealth of studies centered around this in the field of public finance. The basic question, similar to what you have articulated, is who bears how much tax burden? There is much complexity to this question depending on what is being taxed how and who the relevant market participants are.
In your simple example of an excise tax on gas, the question of how much tax burden is borne by the consumers relative to the producers in the gas market depends on their relative price sensitivities. This result is intuitive: for example, if the consumers have their hands tied and would purchase the same amount of gas regardless of price, then firms can simply charge the entire tax burden over to the consumers. In this case, consumers are undeniably worse off, and as you mentioned, it is highly likely that the effect is regressive as poorer consumers would spend more of their income on gas. We could also speculate what ripple effects this tax may have on food, health, childcare, etc.
On the other hand, if consumers are more sensitive to prices, firms will share some burden and collect less revenue, as some consumers would drop out or reduce consumption (e.g., take public transportation) when the price starts to rise due to the tax. Consumers pay higher prices, producers receive lower prices, and there is overall less gas transacted in the market (which results in an inefficiency called the deadweight loss of taxation). It appears that all market participants are worse off, but the public spending funded by the tax revenues may do more good than harm in the end.
Similar intuition using price sensitivities can be more broadly applied to analyze tax incidence in other settings, although technical details will differ. Summaries of some recent empirical evidence on tax incidence are available at https://microeconomicinsights.org/category/public-finance.
Your last point on political pressure to increase corporate income tax rates is especially salient today amid increasing market concentration, inequality, and inter-jurisdictional tax competition for top firms and earners. This a hotly contested topic as it involves equity concerns and inter-jurisdictional coordination (e.g., global minimum corporate tax proposed by the U.S.). You may find some insightful discussion in the last four chapters of "Capital in the Twenty-First Century" by Thomas Piketty and at https://pubs.aeaweb.org/doi/pdfplus/10.1257/jep.34.4.27.