What the modern world has is corporatism, not capitalism, and so profit is the primary motivator. Competition gets in the way of profit by making production and pricing more efficient, so profit-seeking stifles competition and leads to monopoly.
Quite unfortunately there is no way to align the incentives of competition and self-interest; profit and self-interest are naturally strongly aligned.
Self-interest is an inherent human nature. Competition is the mechanism to keep self-interest in check. Everyone wants a monopoly. Our institutions and power structure have to be built to be anti-monopoly of all forms.
Rise of corporatism in America is highly correlated with concentration of power in the state. Corporations could not have achieved their scale and power w/o the state putting its foot on the scale to tilt it in favour of one particular outcome or the other. When you have a giant who can throw its weight around, all you need is a bit of smarts to use that your advantage.
No, mostly not (China has corporatism, Mexico under the PRI one-party regime had corporatism, fascist regimes had corporatism; the modern developed world has capitalism with corporations playing a central role, but thats how capitalism has always been and was a feature of the concrete system for which the term capitalism was coined.)