In context logias are always their element. It is one of the reasons ace kept burning the hats he made for oars. And it's not like that even is an ability that needs to be activated. Hit a logia hard enough without haki and they simply shift. Katakuri in that regard is not different from a logia. He can mold his body like a logia and things go through him like with a logia. Heck, his body can even be split into two like a logias provided haki doesn't cut him.
That said, the haki bit you mention about logia and hardening is not entirely accurate. It makes sense that you can imbue haki into things that have no physical substance so to speak (though smoker can use hardening). However there is no reason magma, ice or sand can't be imbued with haki the same as mochi.
What I am not 100% sure of is whether katakuri can produce mochi to be honest. Mochi is elastic and he can get as much as he wants from his surroundings via awakening. The most likely scenario of course is that he can produce as much mochi as he can like any logia could produce of their element however given how things play out I think there is a very small chance he might be faking that.
Anyways, the reason katakuri is a special paramecia and not a logia is that mochi is not a natural element. Logia literally translates to "nature system" and paramecia to "superhuman system". Katakuri is a paramecia because nature is a bitch to mochi.
i think there is more to it, you may be onto something with katakuti possibly not producing mochi and instead just using his awakening to transform stuff into more mochi for him to use, but even without that there is another fundamental diference between the mochi mochi no mi and logias, although you need to turn your body into your element so attacks pass through you if you are a logia, if your oponent hits your body directly after you have transformed into your element it will still not be hit, unless the attack had armament, but with katauris fruit even if the oponent doesnt have haki an attack that hits his mochi body will in fact hit him, what he does is to not just transform his body into mochi, but modify its shape at the last moment so the attacks dont actually hit the mochi, only glance it, that way he takes no damage, a logia would only need to do that agaisnst an oponent who has haki, like kuzan did with whitebeard, but katakuri needs to do this regardless of wether his oponents have haki or not, provided that the attack is actually capable of harming him that is.
this is actually another jojo reference i believe, some people probably know that katakuri had a jojo reference when he said what luffy would say before luffy said it, wich is a reference to joseph joestar, but this ability would be a reference to the pillar men, when they where awakened the protagonists tried to use hamon against them, hamon being the ability that lets people harm the pilar men in the first place, but one of the things the pilar men did was deform their body out of the way of the attcks, so the hamon infused attacks never even hit them, this is like that, but instead of the pillar men its katakuri, and instead of hamon its haki.
