The original GameBoy went for ~$90 retail when it first launched, which is worth ~$220 once you adjust for inflation from 1989. All things considered, a $250 price point is about on par with what people are willing to pay for a handheld.
Especially once you consider the fact that this is a relatively small business commissioning bespoke parts, and not the gaming powerhouse that Nintendo was when they launched the GameBoy; Nintendo was coming fresh off the massive successes of the NES and SNES. Nintendo had way more negotiating power against manufacturers, because they knew that they’d be selling hundreds of thousands of units, not just a few thousand. Manufacturers would be willing to retool entire production lines for Nintendo.