The thing about CRT pixel "bleed" is that it is a pure Gaussian blur, which is practically the best way of upscaling a raster image within a purely analog system. Strictly speaking, you can of course do much better with a digital-upscaling step, using e.g. Lanczos filtering (which is far better at preserving sharpness while smoothing out "blocky" appearances), or even with special "pixel-art scaling" algorithms that work strictly within the original palette. But still, even a CRT is much better than what can be shown on a fixed-resolution handheld screen!

That's assuming that the pixels are point samples, which isn't true of pixel art. Consider the typewriter on Crono's desk at the start of Chrono Trigger, where a single pixel checkerboard pattern is used to represent the keys. The best way to scale this is nearest-neighbor. The same technique is used in many games to represent chainmail armor.

It's assuming that "a pixel is a point sample" is a better model, even of pixel art, than "a pixel is literally a small square". Checkerboard patterns are also used quite often for simple dithering (in pixel art with smaller-palettes) which is perhaps the polar opposite of your example! (And if you really don't want blurring, most pixel-art-specific scaling algorithms will nicely match nearest neighbor on that particular pattern, while still doing their best to smooth out "jaggies" elsewhere.)

How can the scaling algorithm distinguish between "checkerboard representing high spatial frequency detail" and "checkerboard representing intermediate color"? There are games that use both. The only scaling technique I know that can do this with acceptable accuracy (in real time!) is "human brain scaling".