Freezing rain can occur with a vertical temperature profile completely below freezing, that is to say, without snow being part of the precipitation process. The typical process to get freezing rain is for the snow to fall into a layer of air that is warm enough to melt all of the snow to rain but then fall back into a layer of air that is below freezing before the rain hits the surface. The reason the rain does not turn back into snow is that temperatures near the surface are usually not cold enough for freezing nuclei in the air to activate the freezing process before the rain reaches the ground or an object like trees or power lines.
Temperatures in clouds producing precipitation are usually not cold enough to produce ice without some help. Freezing nuclei are particles that have structures that help ice to form. Some of the more common freezing nuclei are clay particles such as kaolinite. A variety of other dust/dirt particles, sea salt particles, and even some bacteria also function as freezing nuclei. Some freezing nuclei initiate ice formation at temperatures a few degrees Celsius below freezing, but most initiate ice formation at temperatures around -8 Celsius and colder. At temperatures colder than -10 Celsius, enough ice forming nuclei are active to almost assure the presence of ice crystals. At temperatures warmer than -8, ice formation is likely to be limited.
What if the temperatures of clouds producing rain were below 0 degrees Celsius but warmer than -8 degrees Celsius with temperatures below the clouds also below freezing. If no higher clouds are present to drop ice crystals into the top of the rain clouds or the ice crystals from the higher clouds sublimate before reaching the tops of the clouds, then you get freezing rain at the surface. If temperatures are -8 to -10 degrees Celsius, then snow may mix into the rain or change back and forth depending on locally higher cloud tops extending into colder air. A common scenario where this kind of atmospheric condition exists is when the mid-level dry slot of a well developed mid-latitude cyclone impinges from the southwest, south, or southeast into an area of snow. The dry air cuts off the deeper moisture at altitudes with temperatures cold enough for ice formation. If vertical motion is still occurring in the more shallow cloud deck left behind, then freezing rain or freezing drizzle will occur.
The National Weather Service training page titled "Will it rain, sleet, or snow?" gives a brief but informative summary of atmospheric profiles that produce various precipitation types. The section titled "Warm Rain Process" shows an example of the type of temperature profile that would produce freezing rain without ice crystals involved in the process. Note that "warm" in this case means too warm for freezing nuclei to activate ice crystal formation.
"Will it rain, sleet, or snow?" - A National Weather Service training page on winter precipitation types