You believe the 1-category is interesting but somehow not the bicategory, yet one could say the exact same thing one dimension below: Why is this category even interesing while you could just consider the collection of Morita equivalence classes of rings? That is, isomorphism classes in the 1-category you want to consider. Of course, this is because you are probably more familiar with 1-categories than bicategories, but you'll probably agree that's not a good reason! The reasons why the 1-category is more interesting than the plain "set" of isomorphism class are exactly the same as the reasons why the bicategory is even more interesting:
First, the bicategory is there, and it definitely contains information we care about. Morphisms to and from the base field or $\mathbb{Z}$ are the categories of left and right modules, which are definitely object of interest, and composition in this bicategory encodes many operation of interest on modules.
Second, many universal properties in the bicategory don't make sense in the 1-categorical truncation. For example, the group ring and semi-direct product can be seen as quotients by group action in the bicategory ($\mathbb{Z}[G]$ is the quotient of the trivial group action on $\mathbb{Z}$). I need to do some thinking to come up with more example of this, but the bicategory has a lot of the limits and colimits you want in a bicategory (all? I actually don't know), while the category you consider has a lot less. This is analogous to the fact that any kind of universal property in a category won't make any sense if you only look at isomorphism classes of objects.
Many category-theoretic constructions produce interesting results when applied to the bicategory and not when applied to the quotient 1-category. For example, the category of rings is identified with a full subcategory of the slice at $\mathbb{Z}$ of a bicategory of bimodules. I don't think something like this is possible with the 1-categorical quotient.
To summarise, on one hand you have a very well-behaved bicategorical objects which encodes a lot of interesting structure, on the other a much more poorly-behaved 1-category that encodes less information. However, there are some case where this 1-categorical shadow is interesting nonetheless—the idea being that the object we really care about is the bicategory, but some of its properties can be seen on the 1-category.
As you are not familiar with bicategories yet, I'm showing you briefly why the bicategorical limits and colimits or slices are different from the 1-categorical ones.
If you consider a pushout in a bicategory, you have a diagram $A \overset{M}{\leftarrow} B \overset{N}{\rightarrow} C$.
In a bicategory, the correct notion of pushout is the pseudo-pushout (the lax pushout is also interesting of course; I'm claiming correctness in opposition to a "strict" pushout); that is, we want an object $P$ with the following universal property:
For any $D$, the category of $P$-$D$ bimodules is equivalent (naturally in $D$) to the category of triplets of two bimodules $V:A \to P$ and $W:C \to P$ and one isomorphism $\epsilon : V \otimes_A M \simeq W \simeq_B N$ of $A$-$D$-bimodules. Such a ring $P$ can actually be constructed, at least in some cases (I'm not sure about the general cases—I need to think about it).
A pushout in your 1-category would be the same, but you'll only have that an isomorphism class of $P$-$D$ modules would be the same as a pair of iso-classes of bimodules such that there exists an appropriate $\epsilon$, but now the iso-classes are taken on each composent separately without paying attention to the $\epsilon$. This is a very different universal property that is much less likely to have a solution or to lead to an interesting construction.
