I guess modern compilers (meaning anything Arduino era and up, at least when I first got into them maybe mid 2010s) abstract that away, because while true that it's doing that under the hood we at least don't have to worry about it.
Even in the 1980s you had C compilers for micros that had bigger numbers than the platform supported. However, if you've got a small number of registers (esp on the 6502) it's not appealing to generate straightforward register-based code and you're likely to use interpreter methods like
On machines like the Z-80, 6809, and the 8086 you had more real compilers.
The AVR-8 is a special case because it has a huge register file (32 registers!) so if you are using 32-bit numbers you have room for 8 big registers. gcc supports 24-bit integers and those save resources taking just 3 native registers.
