The reason Google font looks like this is because of the setting in the GASP font table. This table toggles grayscale and grid for size ranges. I looked at the Google font sent to your site and for sizes 0-13 grayscale is included. 14-18 for gridfitting, resulting in black and white rendering. Then 19+ is set for both shades of gray and mesh. I bet if you change the size of the text by 13 pixels, it will suddenly become smoothed.
As for Font Squirrel fonts, we set up a GASP table to display grayscale and mesh for all sizes above 4 pixels.
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