Apologies if this theory has already been put forward and I missed it. I am about 2/3rds of the way through Rythm of War currently and have been told a few spoilers so if his reason is directly stated, which I highly doubt, I also apologize.

So we know that at least Rayse and possibly Odium fears Harmony. We also know that this isn't because Harmony is a combined shard, after all he knew that incorporating other shards wouldn't strengthen him, only change him. Theoretically infinite power plus theoretically infinite power is still only theoretically infinite power. We also know that again, despite the fact that their stars are equal in power, Honor and even more so Cultivation fear directly confronting Odium.

My theory is that it is about the immediacy of each shards influence on reality.

Honor and Cultivation are both things of the long term. You could say that Honor is a collection of the past. Habits, beliefs, codes, etc. that are fully displayed and realized only with time. One can commit honorable acts in a single moment, but one can only be honorable with a history.

Cultivation is ones future and the culmination of one's past. You can not immediately cultivate, it takes time. And the more time, the greater the result.

Same with Devotion and Dominion, the other two shards Rayse Splintered, to some degree. While an immediate proof of them might be displayed, their full with can only be displayed with time.

By comparison Harmony, along with the components of Preservation and Ruin, are things of the now. Something is either in Harmony or not. Preservation happens. It's beginning is immediate. It is there now. Ruin is also a beginning, if a beginning of the end. The influence of any of these three will have immediate noticeable affect. Sure it is possible for the affect to build over time, but the change is often most noticable at the beginning and it has a finite end.

Odium I would argue is somewhere in the middle. Rage/passion has immediate noticeable affect, but it takes time for a creature to be primed to have that affect. The greatest influence of rage/passion is immediate, but the underlying conditions take time to build so it might be strengthened. If Honor is the sum of ones past, Odium is the burning of ones past for fuel.

With this in mind, I think Odium fears Harmony because of the immediacy with which he can display his influence. And that immediacy is what most often influences the direct clashes of shards, while the clashes of those influenced by shards most often comes down to the latency/long term affects of a shards power.

It's why Rayse wanted to rush the time between desolations but wants time to prepare against the cosmere. On Roshar he has access to more immediate power. In Scadrial he'd have greater latent power.