Drywall – Is it ok to use joint compound for a thin or skim coat on new drywall

I know this isn't the answer you want to hear, but I don't think priming and painting will hide the tape.

Before you apply any more primer, I'd add another thicker coat of mud. Slather on the mud thick first - it doesn't have to be smooth. Then smooth it out by angling your 12" knife so it's almost parallel with the drywall as you drag it along the surface. You'll have to press fairly firmly in order to avoid air gaps appearing in the thick coat you're smoothing out.

You may have to get a feel for it. If you push too hard and can see the tape again, redo that section until you get it just right.

Wait for the thicker coat to dry, then sand the edges as smooth as you can get them. Very carefully/lightly sand over area covering the tape if you didn't get it quite smooth. You may need to do a second coat along both sides of the thicker coat to make it join the drywall surface smoothly.

Firstly, @Michael Karas answer addresses the "sagging"

Cracking: Premixed Joint Compound (JC) hardens by drying out. As such, it tends to shrink. I only use it for final top coats, where the shrinkage is mitigated and application layers are about 1/8 or so.

Setting JC is dry and mixed with water to a similar consistency as Premixed JC, but it hardens chemically (due to plaster of paris). It shrinks much less.

If your patch is dry and firm, you can continue to overcoat it with premix until you obtain a flat surface.

If it is not firm, remove offending parts until you have a firm base. Replace with setting JC, adding fiberglass mesh tape or other backing as needed. Bare JC shouldn't span more than 3/4 inch without a mechanical backing: Refer to Jon Raynors answer here