I wouldn't buy an Intel 13th-gen CPU, especially not in Europe where electricity has become insanely expensive. If I were you, I'd be looking at an R5-7600. I'll explain why:
The i5-13600K's average gaming performance exceeds that of the R5-7600 by 15%:
However, the average power consumption of the i5-13600K exceeds that of the R5-7600 by a colossal 93% (I'm not kidding):
Not only that, the i5-13600K costs 37% more Danske Kroner than the R5-7600:
AMD R5-7600: 1887kr at ProShop (incl. VAT)
Intel i5-13600K: 2578kr at ProShop (incl. VAT)
The R5-7600 has two other advantages, one small and one large. The small advantage is that it comes with an enclosed air cooler. Now, if you want to throw money away on an AIO, that is your right but if you change your mind, the AMD Wraith Spire is more than adequate for the R5-7600.
The large advantage is the fact that the AM5 platform is brand-new and you'll have access to drop-in CPU upgrades in the future, something that isn't true about Intel's LGA 1700 socket. An upgrade to your i5-13600K will require a new motherboard. This will make what you spend on that MSI MAG Z790 Tomahawk WIFI motherboard a complete waste because you'll just have to turf it if you ever want to upgrade.
For the R5-7600, a great choice in that price class would be:
ASRock X670E Pro RS - 2302kr at ProShop (incl. VAT)
The main PCI-Express x16 GPU slot on an X670E motherboard is PCI-Express v5.0, just like the slot on the MSI MAG Z790 Tomahawk WIFI. As you can see, the difference in gaming performance between the R5-7600 and the R7-7700X is less than 6%, hardly worth the extra 40% (764kr) for a gaming system. That's why I recommend the R5-7600. It's a great gaming CPU, sips power, is far less expensive and is on the superior AM5 platform.
You can do what you want, but I believe that choosing the i5-13600K would be a big mistake at this time.