The problem with that theory is that there is no precipitation there - not right now - no snow fall at all, or rain. It does get frosts there most mornings, and there are mists / thin clouds - but no rain. The only "snow" is carbon dioxide snow at the poles. Dry Ice Snowfall at the Poles of Mars  - It Only Happens on Mars: Carbon Dioxide Snow is Falling on the Red Planet - Universe Today

From time to time over geological timescales as the axis tilt changes and the orbit gets more eccentric as well, the atmosphere does get thick enough for rain or snow. That's not so long ago, on the geological timescale, and will happen again in the future - evidence of gullies carved out by flowing water not long ago but that can't happen right now.

The main competing theories are

• Salt absorbs moisture from the at the top of the flo, as you mentioned. In the press session about the discovery of hydrated salts, they referred to that as a preferred theory but that is just for them, that it is their preferred model. There's no strong evidence for it yet, as you could get hydrated salts with all the models, given that salts are quite common on Mars, which they also said in the press session.
• The water comes from deep underground and the RSLs occur near to geological hot spots. In this case it is replenished indirectly from deep reservoirs, perhaps even from the hydrosphere which may exist kilometers down below the ice layers on Mars (as with Earth it gets hotter the deeper you go and deep down water can be liquid because of the pressure of all the rocks above it so it can't escape - except maybe through plumes that could go all the way to the surface where the RSLs are - it's not necessarily water, it could be sublimating ice migrating up to the surface from the hydrosphere)
• The water comes from deposits at the tops of the slopes that were deposited there millions of years ago by snow or rain. That's a good theory, but the problem here is can you get enough ice up there to supply the RSLs from then on without being replenished again from below?

At present there isn't enough evidence to decide between these theories. The last one is the one most like your idea. There are many papers on the topic, covering all three of these theories.

See also:

Places on Mars to Look for Microbes, Lichens, ...

Water and Brines on Mars: Current Evidence and Implications for MSL

I didn't notice your video. There is ice on Mars, but only at higher latitudes. Some of the RSLs form in the equatorial regions in Valles Marineres. The ground there is dry to some depth.

The problem with ice on Mars in the equatorial regions is that it is not stable. That's because the air is so thin. Water boils at 0C over much of Mars, and at very deep places then it is below boiling point at 0C but only by a few degrees, and ice evaporates given long enough time when the ice is close to or above boiling point. Even at the much colder conditions studied by Phoenix, ice exposed in a trench started to sublimate away over a period of days.

So even if snow did fall it would not be able to remain on the surface to form cornices.

They have searched for ice in the equatorial regions. Have found radar signatures that may be ice below the surface in patches - if so it is probably trapped by an overlying impervious layer to prevent it evaporating.

There is ice on Mars that comes and goes - it's not deposited by snow though, but by water vapour condensing out of the surface. It's like the frost at equatorial regions but lasts for more and more of the day as the winter progresses.

The Many Mysterious Forms of Water on Mars

At lower latitudes then you get frosts in the early morning even in equatorial regions which might be visible from orbit if you took photographs at the right time of day.

But I think the only snow discovered on Mars is Co2 snow. It does have icy clouds, made of ice crystals, but I don't think they ever snow as far as I've heard, not in current Mars.

CRISM Web Site

Though as I said in my answer - at times due to changing axial tilt and more eccentric orbit, then Mars has snow and even rain - good evidence for it. So there could be ancient deposits from those times at the top of the hills for the RSLs. The main question is, if so, how have they lasted for so long without being replenished, producing the RSLs every year?