Hi there
😀
Yeah, stock cooler will be fine if you're not for overclocking. And, Intel will not put anything on the CPU box that is totally unusable hehehe unless it's factory defective

.But, because it is a stock cooler, so do not expect the highest and surprising performance from it.
If you want to have a better performance with the stock cooler, go and buy an aftermarket thermal paste.
But if you want your PC to become quieter and keeps your CPU cooling at a high performance. And also if are planning to overclock at the future, you can go buy an aftermarket cooler
Hope it helps
😀