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GIF can do the same thing much more efficiently than PNG.
Yes it can, but the problem with GIF is color palette. The GIF format only supports 256 colors, whereas PNG has a (virtually) unlimited color palette similar to JPEG. Historically, GIFs are used for logos, clipart and other images where the number of discrete color shades is small, but you need transparency.
Basically, 24-bit PNG is a mix of the best aspects of both GIF and JPEG, but it comes at a price: larger file size.
All that said, from a space utlization standpoint, it makes sense to not support PNG here for uploaded images.
As far as converting to GIF instead of JPEG, I do not recommend it. The larger the number of colors and gradiants in an image, the more stark the difference in GIF vs. JPEG. On some images, converting to GIF won't show any visible difference, but on others you may see banding, muted colors, and other artifacts.
JPEG really is the best choice for stamp/cover images.