Ok so from what you're saying it seems that the standard cooler is really worth it.
Looking over specs I have 3 queries:
1) Why have you picked the x570 plus motherboard? And not upgraded/downgraded?
2) Gamers online seem to be thinking that cpu 6 core min is the requirement for gaming for the next few years and nothing less
3) Having firefox as your main browser?
Standard cooler is fine, after that it's go big or go home haha.
X570 is selected as it allows for future upgrades. The B450 will support the next generation but it's not ideal for it and is going to require a bit of a fudge for compatibility. If you are SURE you won't want to upgrade to the next generation CPU then the B450 is more than enough as it's a great board. The X570 is considering next gen or high end CPUs (3900+)
The cores comment isn't really accurate. It's a lack of understanding IMO. 2 cores are fine for almost every single game going, there are exceptions but they still run absolutely fine on lesser cores.... just not as quick. Where you want more cores is when looking at streaming while gaming (Twitch etc). However, by this point I would often recommend the 8 Core 3800X. The great thing about this entire build is that you can switch between every current CPU between the 3300X and the 3950X with nothing more than a cooler added. If you go with the 3300X just now (very cheap) you won't have lost much if you decide in a year or so to upgrade. However, if you plum for the 3800X and a cooler.... and then don't use it, it's wasted money.
Choose whichever browser you prefer. I'm generally Chrome but I've heard good things about the new Edge model. Firefox is always a trusted browser.