Currently our understanding of biochemistry is so poor that we still have a lot to learn about the very basics. I don't think the diversity of our lab mice is currently a limiting factor.
Our understanding of biochemistry is actually quite good, it is our understanding of biochemistry diversity that is poor (i.e. why are you and I different). Using inbred mice does not help solve this problem.
I'm no expert, but it seems to me that as long as there is still basic research being done on the behavior of very simple proteins, we're a long way from understanding the complex interactions that happen in cells.
Yes we have a lot to learn about all the interactions between proteins in a cell, but this typically does not fall under the heading of biochemistry.
Slightly off topic the traditional divisions between different areas of biological sciences that built up in the 20th C have all broken down with the rise of molecular biology. Once biochemistry was a different field to say microbiology, now both are part of the super-field of molecular sciences.
The important practical results (e.g. many particular drugs) that we get and need are obtained not because of understanding why they work but despite that, empirical testing is a key part of current process.
We have considered and discarded a lot of candidate drugs with great potential before starting human testing. As a rule, if some molecule would cure an important disease but for whatever reason doesn't work on the popular strains of mice, then our current process would cause us to discard it.
