You don't need a high-end cooler like that for a Ryzen 2600.
You would likely be much better off putting that money toward components that will affect gaming performance, such as a higher-end graphics card and a 2x8GB kit of RAM. For optimal CPU performance, you don't want a single stick of RAM, at the very least, and it shouldn't cost much more to move up to a 16GB dual-channel kit of DDR4-3000 or 3200.
Are you in the US, and is this in US dollars? Have you tried selecting your components on PCPartPicker? For an $1100 gaming system, I would definitely want to see more of the budget put toward a better graphics card, and would definitely want to see at least 16GB of RAM in there. For that budget, you could probably even move up to something like a Ryzen 3600. If you want an aftermarket CPU cooler, even 120mm tower coolers under $40, like a Hyper 212 Black or Freezer 34 should be plenty.