Question, assuming that the planet has already cooled down, internally differentiated and formed the pressured ice layer(s): Would meteorites, comets and asteroids provide enough organic and other necessary elements to enrich the waters enough for life to evolve?
Yes, assuming of course the principle source of water on planets is from comets and asteroids. There is some scientific contention around this idea, and although most theories are technically valid; with comets releasing some water as they collide with the surface, they do not do so in sufficient quantities to cover the surface as we see on, for example, Earth. Might that happen in outer-solar-system type environments around other stars? Perhaps so. However, one of the main source of water on almost all terrestrial planets in our solar-system is from colliding with carbonaceous Asteroids and planetismals
that formed roughly where our asteroid belt is. These had rich deposits of hydrogen isotopes deuterium
as well as Aluminum isotropes that can form water via decay. Volcanism and hydrate-mineral dehydration on Earth are also suspects for water-making.
The reason why I brought up the founding of water on Earth in answer to this question is because none of these processes as described above are lacking in organic materials, and unless the superterrestrial worlds have a serious tectonic activity deficiency (which is possible, there are many scientific papers debating the viability of tectonics and crust-dynamics on super-earths, as I trust you are aware, if not, this
are good starts) or have oceans simply too deep (or a planetary mass too great for any viable boundary between liquid and gas), conditions conducive to abiogenesis
should be prevalent sooner or later
(bear in mind, life can be a slow process, although on Earth it seems to have gained a foothold quite fast).
That being said, you seem to specify in your question ice layers, which may be an issue, and their widespread formation might imply that the planet indeed has too great a mass and too deep of an ocean to have much tectonic activity and thus support abiogenesis. Panspermia
by asteroid or dust might be an option, but of course we have no proof that such a method even works. It might if a smaller terrestrial planet further inside the solar-system already spawned life and some of it got up into the exosphere somehow, then transferred by debris to the ocean-world.