You are correct with the Operating System subsystems, but the limitation on running the 16bit programs is down to the Operating System and not the CPU.
A 64bit CPU can still run a 32bit Operating sytem which supports 16bit applications. 🙂
The problem is the manufacturers stop making drivers for the older Operating Systems so you cannot get a new (for example Core Ultra 7) system running fully on Windows XP as no drivers exist for the hardware.
In your situation what I would recommend is running the Pentium 4 in native Windows XP, then running up Dosbox for all you Dos gaming, then you get the best of both worlds without needing to get stuck with native DOS issues.
For a great simple start to using Dosbox with raw DOS files you can check out this short Youtube clip: https://www.youtube.com/watch?v=NIh4GY_TIns
