The Sims 2: Please insert the correct CD-ROM

Solution 1:

Since I legitimately bought these games, I went to EA technical support and told them everything I told here. They asked for my product key, and I gave it.

They proceeded to place The Sims 2 Ultimate Collection in my Origin account. Upon downloading this, I was easily able to run The Sims 2 and all of its expansion packs, stuff packs, and patches.

There are then two known issues remaining: The resolution is locked at 800x600, and Sim shadows indoors are large black rectangles. The good news is that these are well-known and easily fixed.

tl;dr If you have a legitimate copy of The Sims 2 (expansion packs or not), tell EA and they will give you Ultimate Collection, which runs easily on modern computers.