Were you thinking of changing the 4-color palette for every scanline, like the Atari 8-bits and the Amiga could do? Impose arbitrary limitations so you can find clever ways to circumvent them - sounds like fun!
The Game Boy has less memory than would be required for a full frame buffer, so its “just in time rendering” of sprites and tiles is one way to support its full display resolution and still leave some space for game state in the RAM.
This isn't quite true—you need 160 * 144 pixels at 2 bits-per-pixel. This weighs in at 5760 bytes. There are many other reasons why a framebuffer wasn't ideal for the hardware of the time, though.
But in any case a frame buffer would use up almost the entire available VRAM of 8 KB and would have probably also been more expensive since it would have to be dual-port RAM (I believe the Game Boy's VRAM isn't).