IIRC the Lynx was designed at a time (1986-87) when this was feasible for portable systems but not for consoles, because of its low resolution (160 x 102 - the height seems strange until you realize that it gives you almost exactly 16 * 1024 pixels).
> Sega also ended up lifting a lot of the Lynx's design for the Game Gear after the Handy was pitched to them by Epyx.
The external design, maybe, but the internals of the two machines are very different. The Game Gear uses a Z80 to the Lynx's 6502 and has tile-based graphics where the Lynx uses a bitmapped display. The Game Gear is essentially a portable Sega Master System - a console that predates the design of the Lynx.
> There is also an amusing story about how Epyx's pitch to Nintendo of Japan went terribly wrong, though I can't find it now.
It's here: https://www.retrogamer.net/profiles/hardware/atari-lynx/
“We were the first non-Nintendo people to learn of the existence of the Nintendo Game Boy,” Needle says, recoiling even at the memory. “We were crushed. Joe was infuriated. The Nintendo boss left the room and we just sat there, wondering what to do next.”