So this is an interesting one. To start this off, I am running in a virtual machine, so I effectively have a dual-core i5 4690K (4.2GHz) and a GTX 960 with 2GB of VRAM, along with 8GB of system RAM. I am running Win10. So even with these impairments, my hardware is still well above the requirements. The game runs perfectly fine, even mildly modded (mostly visuals & bug fixes), rarely dipping below the 50 FPS mark.
That is, until I enable Bethesda's HD texture pack DLC. Even with no other mods enabled, the same thing still happens; when I load a save, any save at all, after a bit of loading, the screen freezes completely. But it's a bit weirder: the game is actually running fine in the background! I can hear sounds when I open up menus, jump, and walk around. I can also hear NPCs, water, dogs, etc. So the game itself is working fine, it just isn't displaying, and is re-using the previous frame, or not re-drawing at all.
I have tried various config changes, as well as settings, and nothing affects it. According to task manager, the system RAM usage is around 2GB, which is well under my machine's limits. Due to the fact that this is a virtual machine (Linux host) and therefore uses custom video drivers, I can't actually monitor the VRAM usage, but I don't see any reason 2GB would be a limit on a game this old. This also happens with HD texture mods, but not mods that only change a couple of textures (e.g. Verdant Grass).
The only thing I can think of is a limitation of RAM or VRAM, but RAM certainly isn't a problem, as Skyrim is permitted up to 4GB using SKSE tweaks, and if I ran out of VRAM it would just be swapped into system RAM. Sure, it may make the game chug and stutter, but it would still work.