You are mistaken from the first sentence! The purpose of anarchy is to dismantle unjust heirarchies, NOT to remove the existence of authority. It's a common mistake, and one I used to make myself.
I don't think anyone would disagree that capitalism's competitive drive has in ways benefitted humanity, amd I'm grateful for those benefits. It'd be foolish to discard them. Howver, all one needs to do to see how it's harming us more than helping, is to look at the weather.
'Just make a new system' is actually the point of most communist/socialist movements. Because the world is so defined by capitalism, communism right now basically just means 'everything else', but if you look at the history of communism in the world (which is as fascinating as it is heartbreaking) you can see that there is a huge variety in beliefs of how it should be instituted! Socialists, anarchists, tankies, environmentalists (yes, ecology is intend3d as that alternative system you suggest) all have to act as more or less the same movement because of the overwhelming success of capitalist propoganda
If capitalism truly was a level playing field, free of inheritance and prejudice... I would certainly agree with you. But it too, is imperfect. I would encourage you to compare, sometime, the results of a country being permitted to attempt communism. Modern day Vietnam is a lovely place, for example.
Regardless! The fact that you seek socialism makes us Comrades, because I do too. I'm actually a pacifist! While I have pretty radical beliefs, I have peaceful hopes for achieving them. Which means a road back through democracy, into social democracy, and hopefully one day direct democracy. We both want the same thing, less greed. I'm happy to fight (metaphorically) side-by-side with anyone who has that same goal.