That issue aside, and getting back to the issue as to whether one would go with the 12400 or the 5600X.... TL;DR - you can't go wrong with either one, when the MB+CPU price is close.
My more long-winded explanation:
The motherboards between the two platforms more or less start, price-wise, in the same area, give or take. Maybe there's a slight edge for the AMD platform being slightly cheaper.
The 12400/12400F, while the official prices are in the $170-200 range, seem to have been hovering in the $200-210 range (based on PCPartPicker), though I could've sworn I saw a 12400F in the $170-ish range yesterday. That said, even in the $200-210 range, it's got an advantage over the 5600X in price, that the lower MB prices for AMD don't completely make up for.
When I was poking around recently considering a budget system, I was looking at the 5600G, but then the 12400 came in. If it wasn't for the proximity of MicroCenter, and, the timing of their deals, coinciding when my searches on the 12400 and Intel motherboard resulted in higher prices, I might've gone the Intel route.
Today, for me, even though I lean in favor of AMD since Ryzen, and also did back in the days when Athlon ran over the Pentium 4 around the turn of the century, my purchase might've been the 12400. But, the fact that the 5600X and its platform beat it price-wise on that week, and the still better power efficiency of the AMD appealed to me.
The 12400 power efficiency isn't terrible, but it still can't touch AMD. So, I'd generally give AMD a few dollars leeway.
That said, right now, if the prices are close, the choice is almost a coin-toss. The problem is, the big online vendors aren't reducing the 5600X's price very much yet. They're going to have to, otherwise they'll be sitting on those CPUs. I suspect that where MicroCenter goes, Newegg, Amazon, etc., will have to follow, even if more slowly/begrudgingly. MicroCenter alone isn't going to move them, but MicroCenter, plus the pricing pressure from Alder Lake will.