Corporations are an ancient concept. They were widespread in Rome, the concept being that when a group of people band together they form a body (corpus) that is treated as a separate, independent entity. The Romans allowed corporation status for burial clubs, craftsman guilds, and religious cults, but also for state and municipal governments.
This deep history of corporations in Western civilization continued through to America -- although the US constitution doesn't specifically mention them at all, their existence was endemic throughout the colonial period of US history. The original corporate concept during the colonial period, a "charter", was usually granted by the king or parliament, and often also conferred on the chartered corporation a monopoly. As such, the corporation was more correctly seen as a special form of political patronage or reward.
Since the 1818 case Trustees of Dartmouth College v. Woodward, the US supreme court has considered corporations to have the same rights as natural persons, and since the 1886 case of Santa Clara v. Southern Pacific the courts have applied the 14th amendment equal protection clause to corporations as well (another reason why that amendment has been such a well intentioned but unmitigated disaster for American society.)
There are many reasons trotted out as to why the corporation is a bedrock to the economic structure of Western civilization, but they boil down to stability of the entity despite changes to personnel, ease of paperwork for unrelated citizens to invest in the corporation, and protection for owners and investors from liability. Personnel changes would be a very small and easily surmountable bump with existing laws if all corporations were instead partnerships, and investment in partnerships can already be accomplished through structures like the master limited partnership. Thus the main reason why corporate personhood is vigorously defended and considered "vital" is because of its protection from liability. Defenders of corporate personhood assert that if this liability protection were to vanish, no one would be willing to take any risk, and the engine of innovation would sputter and stall.
That is an interesting argument, because it begs the question -- does the concept of corporation actually reduce "risk"? Or does it simply reassign it?
Let's consider an imaginary corporation we can call the Thomas Company (TC.) They propose to bore a tunnel through the peak of Mount Impassable, which will cut transit time between the two valleys on either side in half. This is a dangerous project, mostly because Mount Impassable is unstable, and boring this tunnel could start a landslide that ends up destroying the farmland in either valley. This risk of damage has so far precluded any individual from being willing to attempt the project. But -- enter the Corporation!
Because the owners and investors of TC are shielded from liability, the TC is able to raise funds and start the project, and the only risk the investors are taking is the money they put in, which they could theoretically lose all of, but their risk is capped at this amount. None of the rest of their savings or net worth is at risk. The owners of The TC also stand to lose any invested money and certainly all of the time that they put into the project, but their risk is also capped at this amount.
This capping of risk acts like a one way valve or barbs on a hook for society. It encourages movement in one direction, towards investment and the attempting of projects, and it does this by providing a floor to the downside of taking risk, but no limits to the upside of taking risk. Hence its "indispensability" in encouraging the continued taking of risk on the part of society.
So let's walk this through. Imagine that the boring of the tunnel is successful. Society benefits in that transit time is shorter, the farmers in the valleys on either side pay less for getting their products to market, and the TC investors make out like bandits. Everyone wins! Of course, the farmers do have to pay to use the tunnel, which is how the investors recoup the cost to bore the tunnel and make their profit, so the farmers and society in all reality are actually paying for the tunnel, they just do it over time, plus they end up paying the investors and TC owners profit, but the total cost to them is still less than what they used to have to pay to truck their stuff around the mountain. So again -- a win for all of society! The time and energy spent to build a more complex solution has paid off.
But let's look at the other possibility. They try to bore the tunnel, and a huge landslide covers much of the farmland in either valley. The gamble failed. The investors lose their money, the TC owners lose their time and money, but thankfully the liability is "capped", so society doesn't lose. But wait. The farmers lost. They lost their farmland. They lost everything they had. And the company went bankrupt, so there is no money to repay the farmers. They just get shafted. The flip side to the liability being capped on the part of the investors and owners is that other people, people not involved with the TC company in any way, will lose all of their money so that the investors and company owners don't have to. The corporation structure didn't actually "limit" the risk of landslide. It just passed the cost of a landslide on to someone other than the investors and owners.
So what does the whole concept of corporations limiting liability actually represent? It just represents the investors and owners stealing the risk reward from the farmers. Any investment of time and energy to create a more complex solution to a problem runs risk, which is the risk not just of the project failing, but also of damage to the capital of society. Before anyone embarks on a risk, they assess whether the potential loss of time and energy as well as the potential damage to their stuff is worth the benefit they will get if the risk is successful. This potential for loss is what makes us all cautious. Which is a good thing.
A corporation, by reducing the risk to owners and investors, makes society more willing to attempt risky things. But this risk is not actually "reduced." It is just passed on to the rest of society, who also become unwitting investors in the project. They are not consulted, and they gain little of the rewards if the project is successful. But they bear the risk if the project goes tits up.
In our imaginary scenario, the TC undertakes risk for the farmers, owners, and investors. The farmers are just not consulted first. The farmers get very little of the benefit if the project succeeds -- the investors and owners keep for themselves the costs to pay for the project as well as the profits, all of which come out of the farmers pockets, which means that the farmers actually paid for that project as well as for the profits. And if the project fails, the farmers bear most of the costs. And the concept of the corporation, by very nature of the limits on liability so vaunted by its defenders, is the exact same concept that steals the reward for taking this risk from the farmers and gives it to the TC investors and owners.
Corporations are simply vehicles for re-apportioning risk and reward in society. They don't change the amount of risk at all. They just move the risk and reward from one set of pockets to another.