The GIF format is not lossy. There is no "quality" option in the format itself; just a color depth option, same as PNG.

Photoshop uses the same tricks for "lossy" gif that this article uses for PNG.

I've researched lossy GIF too: http://pornel.net/lossygif but even lossy version wasn't better than PNG.

LZW is just a very poor compression. Even best case is taking ridiculous amount of bits (you can only add 1 byte to previously used pattern, so it's sequence of symbols for 1+2+3+5+6+etc. pixels resetting every 4k iterations), so there isn't even room for improvement.

So, GIF is awful. Should be forgotten.

Only reason GIF is still alive is its animation support. APNG and WEBP+animation are not there yet.

8-bit palette and crappy compression do not matter when you offer exclusive features.