In order:
- If I had to chose between the two, with no plans for future upgrades, I would pick the 13600K. It will arguably age better than the 7600X while offering better performance today.
- If I took future upgrades into consideration, then I would pick the 7600X for the AM5 platform and the option to upgrade to the next gen products.
- Last but not least, if I took pricing into consideration, then the equivalent in price to the 13600K would be the 7700X. I would argue the 7700X is just as good for gaming as 13600K, and I would pick 8P over 6P+8E in gaming any day. (for productivity it would be the other way around)
Conclusion: the 13600K is the better all-purpose CPU with no plans for upgrades (can see this is a choice, not necessarily as a limitation). The 7600X is the cheaper
good enough alternative with money saved towards a future upgrade. The 7700X is the better gaming CPU today within the budget, with options for cheap drop-in upgrades in a few years from now (probably when you upgrade your GPU, no sense until then).