Why don't SSDs use the +12V rail for programming/erasing?
Three reasons really - TLDR: Reliability, and Compatibility


SSDs use 1.8V in their logic circuits to prevent interference.


AFAIK, The 12V level for programming and erasing applies to NOR flash cells, I'm not certain what the voltage level for NAND and VNAND flash is, as they use a different method to program and erase data.  (Why does NAND erase only at block-level and not page level?).  The voltage MUST be higher than Vccq, but not necessarily 12V.


Flash Cells are unreliable, and must use a controller to be usable.


An SSD is just Flash Cells with a controller that emulates a hard drive.  By directly piping in a 12V line past the controller to the logic side of the device, you're really playing with fire.  Flash memory is basically designed to fail predictably, and the controller is always maintaining the device, cleaning up messes, and sanitizing data streams. While the proper voltage rail could be supplied from the power input, it's more reliable to use localized voltage regulation within the controller's circuit space.


and finally, even modern computers don't have reliable voltage rails


The motherboard has a lot of voltage regulators itself because of this to provide clean power levels to the more sensitive components of the motherboard.  Many PSUs have a limited number of 12V rails, and quite often on cheaper models there will be only 1.  This can cause voltage dips as well as surges, the most I've seen in practice is 12V ±1V, but flash cells are extremely sensitive to voltage levels, and relying on external power regulation could end in disaster should something jump on/off the same rail and pull the voltage up or down.  This would actually be common since an erase or write operations mean the computer is busy and will be pulling additional power compared to idle.