The current intel platforms are more mature and more likely to work out of the box.

The AMD platforms provide stronger raw performance and a much better future upgrade path but with more potential for system instability and very slow boot times.

So just go with whichever you think will be more important to you.

3060TI should be plenty for 1080P, but if your not happy with the performance move up to the 4070.

Any single tower cooler should be fine.

The AK400 or Thermalright Assasin X120 Refined SE are two popular choices.

Pentium 4 > AMD X2 250 > i5 4690 > i5 13500

A 12100/13100/12400 with 16 to 32GB DDR4 Ram and a cheaper B660/B760 motherboard should do the trick (assuming you are recycling the rest of the parts).

In games where you will be GPU bound (which will be most games if you don’t have like a 4080 or 7900XTX) these two CPUs will offer almost identical performance.

In games where you a CPU bound, the performance difference will be like 5%

The 13400 is not a very big upgrade over the 12400 and not worth it for the extra premium.

Where as the 13500 has increased cache and is like 15% faster than the 12400 and the 13600K would be about 25% faster.

So my pick would be the 12400F as the intel 6 core budget king, or 13500/13600K if you had the extra money.