I agree. With both intel or amd, I need a new motherboard. Intel's b660 are currently selling for almost half the price than its "budget" b650 counterpart, and I can use my DDR4 RAM with intel.
It boils down to being about 300€ cheaper going with the 13600k.
5800X3d although selling for almost 500€ here, is still the cheapest solution for my current AM4, albeit not the best for a mixed gaming/productivity stand point and no upgradability whatsoever down the road, nor any tunability.