Animated GIF creators using Photoshop are in kind of a dilemma right now.
Save for Web is very old code, so old that certian operations such as editing an animated GIF can be very slow, and there are some size limits that people run into.
Adobe has expressed an intention to start over, so they created Export As, as a modernized potential replacement of Save for Web. That’s why they renamed it Save for Web (Legacy), because the idea is that old Save for Web is going to go away instead of being updated. But the problem with that is, like Stephen_A_Marsh said, Export As doesn’t handle animated GIF (yet?).
In the meantime, Photoshop added an updated GIF format option to File > Save a Copy. If the Photoshop document has an animation timeline, this method will preserve the animation. But (again), you only have a few options, so few that you don’t get to tweak the GIF color palette.
And as you found, Adobe Media Encoder doesn’t offer many options either.
So that’s several options, but none of them is a single complete and effective GIF optimization solution, which means…
If I have a challenging animated GIF, like one containing many colors, I will export it as a video, then run it through the free software Gifski. I get great GIF animation results that way, the file can be adjusted for smaller document size, and their advanced color optimization algorithms make it look great.
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