Honestly, to mimic consoles of yor, you don't even really need anything as complex as OpenGL (though its certainly an option).

The straight-forward approach you describe will also only be good for simple tile-based games, and won't really recreate the console experience. Alot of that console magic had to do with the use of palettes, and with manipulating the hardware between raster lines. You'll have to emulate those things too if you really want that olde time experience. Most of that stuff is easy enough with a software renderer, but utilising GL or D3D draw calls complicates things like per-scanline effects (still doable, of course).

Still more of the console feel is around hardware limitations, so you'll have to decide whether you want a truly realistic approach, or something that's more like an old console with unlimited but largely static resources.

But we don't have square screens anymore. You'd have to center it over a background image, or use a 16:9 or 16:10 resolution in the original image. Such as 320x180 (16:9).

I found 384×240 (16:10) and 384×216 (16:9) to be better options.

You probably mean integer multiples instead of powers of two (e.g. 3 is OK, 2.75 is not). And SDL defaults to no texture filtering, for the record. (EDIT: wait, I think you mean upscaling the textures - upscaling the coordinates of the triangles should be enough, no need to change the textures)

The problem is that then you risk the coordinates to not be aligned to the fake (scaled) pixels, which completely ruins it (this is probably the biggest issue with pixelart on modern games). You'll need to take this into account and modify the coordinates to make sure they always end up aligned to the scaled grid.

Note the above workaround only works if you're drawing the sprite as-is and you limit special effects to just flipping (things like coloring aside). If you add scaling or rotation, you're pretty much guaranteed to break the alignment (though those effects on pixelart don't look very good anyway, faking them with hand-drawn sprites is usually better).