Don't bother with the paste, AS5 isn't exactly a 'good' paste to start with, it's very mediocre in performance and requires periodic replacement. Should never be used on an AIO because it's a 'dryout' paste (which is fine) but once it dries out it becomes susceptible to vibrations from AIO pumps. There are much better pastes for AIO use.
The cooler is overkill, for such small wattage cpus, but that's fine, just means the fans will never need revving up to anything close to full speeds, so noise is at an absolute minimum. You'll get the same performance out of a smaller AIO, a 240mm is basically the same as a Big Air cooler, which is still overkill. The only advantage offered by a 280mm/360mm AIO on that cpu or its equivalent is the fact it comes with case fans, which generally requires a seperate purchase when using air coolers.
Air or liquid do the exact same job, they just go about doing it in a different way, have different advantages and drawbacks. Cost is neither. The air cooler by itself is cheaper, but 3x pack of rgb case fans can run as high as $120, so an air cooler + case fans can easily cost more than an AIO, but the advanced AIO's with fancy OLED displays offer advantages to some, but can easily cost more than an air cooler + case fans.
So do the research to figure out exactly what you Do need, then decide what extras you want or prefer, what are advantages or disadvantages to you personally. On a 12600k/13400, air cooling is most definitely a viable option.