At the $1$-categorical level we can 'demote' a category to a set by letting it be discrete, and every category has a canonical discrete subcategory that we can view as its 'demotion' to a set given by all the objects and only identity arrows.

Is there a similar notion of 'demotion' for bicategories to $1$-categories?

We can define a 'discrete' bicategory to be a bicategory with only identity $2$-cells, but this immediately forces it to be strict since certain $2$-cells are part of the defining data of a bicategory; if the associators and unitors are identities then horizontal composition commutes on the nose and $1$-cell identities vanish under composition.

This means that $2$-categories (strict bicategories) have canonical demotions to $1$-categories given by the above definition, but bicategories don't.

Specifically, this prevents us from having a canonical discrete sub-$2$-category $\mathcal{C}$ (as defined above) for an arbitrary bicategory $\mathfrak{C}$ since the unitors and associators in $\mathfrak{C}$ may not be identities, so we'll have to choose new composition/identity functors in $\mathcal{C}$. I believe that the new functors will be a postcomposition of the previous composition/identity functors with the isomorphism quotient mapping $q:1-cell_\mathfrak{C}\to[1-cell_\mathfrak{C}]$, but this means that we have no canonical embedding back since (for example) $f\circ(g\circ h)$ and $(f\circ g)\circ h$ will be equal in $\mathcal{C}$ but not (in general) equal in $\mathfrak{C}$.

Is there some other, more well-behaved definition of a 'discrete' bicategory such that all bicategories have a canonical discrete sub-bicategory we can view as it's demotion to a $1$-category? If not, is this because the 'weakness' involved in the definition of a bicategory is not 'visible' at the $1$-categorical level?

We might be able to circumvent these issues at the $2$-categorical level by appealing to a coherence theorem and obtaining an equivalent strict $2$-category, then taking the discrete sub-$2$-category of the strictification, but this answer is less interesting to me because full strictification stops working at the $2$-categorical level.

Is there some notion of a 'discrete' weak $n$-category in the literature such that all weak $n$-categories have canonical discrete sub-$n$-categories we can view as their 'demotion' to an $n-1$-category? If not, is this somehow because the new 'weakness' that appears at each level is invisible from the previous level?

It seems like the step from $2$- to $3$-categories actually introduces some fundamental new weakness that stepping from $1$- to $2$-categories doesn't, so an answer for $3$-categories (perhaps involving Gray sub-$3$-categories?) may suffice to answer both questions.