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So the common explanation (except for raw gravity) is that there not enough air pressure at the top edges of the atmosphere to push the molecules/atoms out into the vacuum, since that's actually what's happening (vacuums don't actually "suck" from a physics standpoint), but this is just kicking the can down the road, because it doesn't explain why the entropy does not cause the more dense (higher pressure) molecules to push into the less dense areas, and the only plausible explanation for this is that this is all taking place within a container with a finite volume.

If we are in a sealed container of finite volume, pressure should be equal, regardless of altitude (see Pascal's Law). Either Pascal's Law is incorrect (which it doesn't seem to be) or we're missing some factor(s).

You know about inches of water and inches of mercury as a measurement of pressure, and that as you go deeper under water in a lake, or wherever, pressure increases. Why this pressure increase?

That pressure obviously due to the "weight" of matter above the point.  Billions of gallons of water are pretty heavy.

But with a vacuum on top the "weight" of matter will get lighter and lighter as the upper layers press out into the vacuum of space, and that would continue until the liquids themselves are turned into gas and eventually evacuated.

It all requires a container and cannot happen without a container.

This should be demonstrable on a small scale if gravity can produce this effect.  Put some dirt into a chamber, perhaps a cup of water here or there.  The introduce gases in the same mixture / proportions that we would have in our atmosphere, and fill the remaining space to within a reasonable average PSI of the earth's atmosphere.  Over time, not sure how long, but we should see the heaver gases lower down, closer to the dirt and the lighter gases higher up.  Not sure how long it would take to settle out that way.  But whether you put a tiny amount of gas in (tiny air pressure) or a larger amount (high air pressure), introduce a vacuum to the open top of the chamber and not only will the gases dissipate, but the water will evaporated (turning directly into gas) and it too will spread out to fill the available vacuum area.  Gravity does not suffice to keep those gases nor the water in the lower chamber, nor does the explanation of lower gas pressure at the upper reaches of the atmosphere.  Even with tiny air pressure, very quickly the water will begin to transform into gas and will then evacuate the lower chamber.  This debunks the explanation that a combination of gravity and the low air pressure at higher altitudes explains why the the earth's atmosphere and oceans do not evacuate the earth out into space.

You could argue that there's some unknown principle or force keeping things bound to the earth, but modern science has articulated no such explanation.

Some of us accept the unanimous Patristic interpretation of Sacred Scripture that there is a solid firmament above the atmosphere that keeps it contained.  Others prefer to bow to modern science.  Science is proven wrong over and over and over again, so much so that only a fool would put much trust in them, especially when you see how the origins of modern science are clearly driven by the atheistic agenda.  They assured everyone that the Big Bang was fact for several generations now, that the universe is expanding, that gravity is the primary force that governs the movements of the galaxies, stars, and planets, but this has been so badly debunked that they had to make up this phantom "dark matter", which allegedly must constitute 80% of the mass of the entire universe to keep their theories on life support, even though no one has ever detected this dark matter.

Pressure and density are two separate and distinct things.

Yes, but they're directly related and, depending on the context of what's being discussed, can be used interchangeably:

Pressure is the measure of force acting on a unit area. Density is the measure of how closely any given entity is packed, or it is the ratio of the mass of the entity to its volume. The relation between pressure and density is direct. Change in pressure will be reflected in a change in density and vice-versa.