You should add an ambient freezing effect to cold, snowy/icy biomes as a challenge/adventure to the player's explorations. The following would help the player counteract the effects of the cold:
-Can be crafted from the furs of animals and any variant of armor (except dyed leather & chainmail). 1 fur is required for a helmet, 6 for a chestplate, 4 for leggings, and 2 for boots. Worn to slow down the freezing prior to damage (see further below). A full set of fur armor will eliminate the freezing effect completely. The armor's durability and protection from regular damage is equal to the regular armor variants.
- Polar Bear fur (gives the mob more purpose, but does not increase durability or freezing resistance)
- Llama fur
- Rabbit Hide
Another Armor Enchantment (an alternative to Fur Armor): Insulation:
-This freezing-resistance alternative is an argument for those who may disagree on adding fur-trimmed variants to the game's armor. Like an opposite to the Frost Walker enchantment, Insulation could be an enchantment applicable to any piece of armor that resists the freezing effect. Not wearing a full set of armor where all armorpieces carry the enchantment will only slow the speed of the freezing, but a full set will eliminate the effect completely.
-Will provide protective warmth in a radius based on the block/block entity (blocks are listed above in level of effectiveness)
What the Freezing Effect Does:
-When under the freezing effect, the player's hp bar will start to turn icy blue from the left to the right. The different tiers/levels of this effect determine the speed of the ice creeping along the row. Once the entire row is "frozen over", the player will take damage until death, until a full set of fur clothes is worn, or until a heat source is reached. The player only takes damage when the whole row is frozen, not as it is freezing or partially frozen. The point where the whole row is frozen is when the player is too cold.
-In line with heat source blocks and fur armor, sprinting (double w or ctrl+w) could perhaps be a way to slow down the player's hp bar from freezing over due to sped up body temperature, the cost being their hunger bar depleting more rapidly. (https://minecraft.gamepedia.com/Sprinting)
For those who don't want cold biomes to fully change any favored mode of play in survival, the menu's difficulty option could turn off the cold biomes' influence of automatically freezing players on Peaceful. Easy, Normal, and Hard could bring different levels/speeds of the freezing effect: levels 0, 1, and 2 respectively. The following chart shows when these levels of freezing effects could be reached:
*Note that this chart's listings are all conditioned in snowy biomes with snowy weather. Other coolish biomes could be incorporated if reasonably possible.
Freezing as a Potion Effect:
-If useful as a potion or splash potion, a Potion of Freezing could be brewed the same way you brew a Potion of Weakness (with a Fermented Spider Eye), if the Brewing Stand is placed on Packed Ice. The results on a mob would be like a delayed wither effect. The freezing over of the hp would be timed on how much max health the mob has, so long as the potion's effect lasts longer than the freezing can cover. This could make an interesting additional mechanic to maps.
-Extra: for better integration and continuity, the Stray skeleton's arrows of slowness could be replaced with arrows of freezing, or perhaps giving them a mix or random chance of either arrow, making them a more interesting variant.
*The Freezing effect inflicted by the frozen biome itself will only effect players, not mobs, naturally spawning and command-generated. Mobs could however be affected by the freezing effect via potions, tipped arrows, or /effect.