A major complaint I have is that the item shares a similar aspect of when to build it as Titanic Hydra. Both have effects you'll typically be using to waveclear, which while it is a problem as a top laner who doesn't build Tiamat (aka most juggernauts), it'd make more sense if this was another Tiamat purchase with the current effect. Then there's the build timing.
The gold worth and stats themselves can be debated, but the "when is this built" is unsolved. Frozen Mallet on Gnar, Teemo, etc. are first or second item choices. Urgot prefers to build it only after his Cleaver so that he can make use of the reduction. This overlaps with the first Phage purchase and Phage speed-up effects, which have diminishing returns because of how movement speed works.
Now, onto a side note here -- Kindred and Kog'Maw, hell even Kalista at times build Frozen Mallet to die much less quickly and add higher team utility. They need their damage items first before they can ever consider building it, with stuff like Kindred being able to build it second item. Shyvana has become overly reliant on this item in comparison.
I like the design greatly like the Lethality changes where it's based on the opponent's stats and not just an overtuned effect. But the item simply would not replace the current one for melees (based on what you've given) and would still be greatly in favor to ranged champions as they all have lower MS than melee characters (with things like Teemo being the exception after he gets levels into W). Even despite that, it'd never be built.
I suggest making it so that Frozen Mallet actually gets tuned power to the user in a more windowed fashion, which is the problem we've had with this item consistently. Active effect that makes it so the first auto-attack slows the enemy after a channel (so that the enemy can see it) with a strong slow effect that is comparable for both melee and ranged for a short duration. The cooldown would be still relatively short to make use of it in teamfights multiple times (shares current active use on champions that would rely on current Mallet) but would give the opponent actual breathing room against the item. As for tuning the numbers or if it'd need ambient resets or anything like that would remain to be seen, as this is just brainstorming.
Again, great concept you've given as it sparks discussion. Hopefully we see more simplistic ideas for implementation that solve what has been a long term complex issue, because those tend to be the better solutions instead of the super roundabount changes we've seen before on the PBE.
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The active isn't very ideal for other ranged champions who currently use Frozen Mallet for perma-kiting 
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If many ranged champions still used it, the passive/stats could be nerfed and the active could be buffed.
The Cold Rage passive wouldn't necessarily need to be unique from regular Rage passive on phage items, since the slow is greatly reduced and juggernauts/bruisers already get kited pretty hard.
seems like it will be obscene, especially considering the base movespeed from trinity and the fact that the phage proc will still work.
interaction.
Yess i like this idea of oneshotting squishies
