No, you've described the caricature of capitalism that those opposed to it repeat to themselves to help justify their opposition. The central philosophy of capitalism is that voluntary trade leads, ultimately, to greater and more efficient production. That's not to say it's necessarily a straight line walk or that what is produced is necessarily pretty. Competition is a side effect that comes into play once you are producing something valuable to someone else willing to buy that something. It is a mechanism that keeps check on the bounds of prices and efficiency.

The real difference between capitalism and socialism is that capitalism requires voluntary action whereas socialism embraces compelled action. Socialist "cooperation" and "coming together to solve problems" are euphemisms for a politically powerful group to harness the productive output of the governed for its own purposes, without regard to if the perceived problems or the perceived solutions are real or not. Unless, of course, true socialist policy allows dissenters to opt-out of participation, in which case I withdraw my characterization. Assuming I'm right, I find the notion of "cooperate or else" far less appealing or moral than capitalism with all of the competition that comes with it.

Finally socialism doesn't eliminate competition... it only moves it from the marketplace to the seat of government. There is no grand collective, there is no "public good": there are individuals with differing beliefs, interests, and objectives. Once you put sufficient power in the government: the most ambitious individuals will compete for privileged positions within political movements rather than businesses, political movements will compete with alternative political movements for the reigns of government. Socialism doesn't remove fundamental human desires for prestige and power, it only redirects the energy into the mechanisms of state; and no that's not necessarily a benefit to the poor, or the underprivileged, or any other banner of victimization that you might fly to justify that state. The inability of a government program to fail means that the real world effectiveness any given government program (assuming that it's even a "right program") is secondary to its ability to attract political support. In capitalism, the voluntary aspect allows for failure if the business isn't producing the product efficiently or it isn't producing something that someone else actually wants or needs.