Can you be more specific of what "properly manage a palette" means?
Essentially how a lot of things with regards to frames and palettes work in Aseprite. Click and drag to select, click and drag the yellow outline to move. Edit (Guess surt already said this, whoops)
previewing ... frames not possible

The above is beta only. The beta is quite stable, though. If you've got Aseprite on steam, do this:
https://www.aseprite.org/beta/If not, ask for a Steam key.
naming ... layers is not possible

You got me on naming frames. You can add user data to cels, but it's not helpful for what you probably use it for in GG.
moving multiple frames at once seems to be not possible.

It's possible if the frames are contiguous or non contiguous.
You can also link non contigous cels (but only on the same layer) which seems to slip under a lot of people's radar.

Essentially in a layer in "continuous" mode, control+click and drag creates a linked cel. In a "discontinuous" layer, that will copy.
I don't know if I can merge/copy colors

Control+click to copy.
I don't know quite what merge means. Give a specific use case? In the very worst case, you can quickly replace a color across all layers and frames. (If you wanted to make say... index 8 use index 0.)
Edit: Hmm... actually, it probably means merge duplicate colors. Like if you have two of the same red in the palette. If that's what you mean I can think of a way to do it, but a better way should exist. Confirm that's what you want and I'll open an issue on github.
if you want to adjust a color you actually have to open the palette editor.
You can freely add colors to the current Aseprite palette with one click. How does a program that handles "adjust a color" do it better, though? Because the Aseprite dev is pretty open to suggestions, you could probably get the behavior you want.
This may sound a bit snarky, but looking for a new program (Pro Motion), and then looking through the help of that new program without looking through the help of the first program (Aseprite) is puzzling to me. There are some things about Aseprite that aren't the most intuitive, but Pro Motion is
really not the route to go for intuitive.
If you give more specific GG actions you want to do, I can probably tell you how to do them in Aseprite. Just... be specific, because I don't use much Gale. Know Aseprite is a lot more powerful than people seem to think!