Normal ordering is pretty useful to stop expressions from diverging in quantum field theory and works out perfectly fine regarding this, but there is this little problem: Consider for example an annihilation operator $\widehat{a}$ and the corresponding creation operator $\widehat{a}^\dagger$ with their canonical commutator: $$[\widehat{a},\widehat{a}^\dagger] =i\hbar.$$ Applying normal ordering (and reordering $\widehat{a}\widehat{a}^\dagger$ into $\widehat{a}^\dagger\widehat{a}$) makes the left side vanish, while the right side doesn't change, resulting in a contradiction. In general, having a longer expression of operators, swapping an annihilation with a creation operator using this commutator, knowing it won't change their normal ordering at all, would just create a term with a $\delta$ distribution out of nowhere.
It therefore seems like normal ordering can map same inputs to different outputs (which is horror for any mathematician) and therefore the operators already have to be in a very specific order before normal ordering can be applied, if there should be no arbitariness in the result.
Since post like here (commutators are forbidden inside normal ordering) and here (proper definition of normal ordering) already give an answer to this specific problem and I could only use more and maybe different clarification concerning it, the focus is on the following questions, on which I haven't yet found anything:
Is normal ordering indeed arbitary? Is the arbitrariness arising here maybe exactly why it is able to prevent diverging expressions in the first place? What are the possible changes of its definition to prevent this and more importantly how can they be justified? Are there motivations other than to just avoid contradictions? Since the exact same also holds for its introduction to avoid divergence in the first place: Is there a suitable physical interpretation for using it?