Have you been able to buy a 3060ti?
If so, great. It is the graphics card that is most important for games.
If you are budget constrained, and you want to buy today, look at the $125 intel i3-10100 as a budget gamer.
Here is a review:
Intel's Core i3-10100 is the most affordable Comet Lake "Core" processor. Unlike its predecessor, it finally has HyperThreading, which brings the core configuration to 4c/8t. Our Core i3-10100 review takes a close look at how Intel's new budget offering performs against AMD Ryzen 3 3100 and 3300X.
www.techpowerup.com
It seems competitive with the 3600.
It includes a perfectly adequate cpu cooler.
Ram speed is limited to 2666, and ram speed will make no difference to intel anyway.
A MATX B460 based motherboard will be $80 or so.
On a budget, I think intel 665P m.2 nvme is a good choice.
You can always add storage later.
That said, if you can wait a bit, intel rocket lake is about to launch.
No doubt the K suffix processors with a 19% IPC boost will get all the press.
But, there will be lesser processors without overclocking capabilities that should be of interest to a budget builder.