I heartily don't think we should and this is why:
the Catechism is a sure norm for teaching the faith. The Catechism has thousands of citations, and only 28 of them are from canon law.
It makes answers seem legalistic instead of doctrinal (or at the last friendly). It shows a one-dimensional view of Catholicism, which nearly no one maintains.
It often "cause jumps" to solve a persons problem by making assumptions
It is not a "sure norm" for teaching the faith like the Catechism is. The Catholic Church herself probably doesn't want us to use it without the proper faculties.
No one teaches or learns Canon law until graduate studies. AFAIK few priests even know a whole lot about it.
It is too easy to search and find an answer in it. There are thousands of better older off-line sources. It is a lazy way to get a primary source.
It should not be the go to answer to "What does the Catholic Church say about X", that should be Magesterial Teaching, Tradition and Scripture (i.e. just because it's not linked on Vatican.va, doesn't mean it's not official)
It almost always must be prefixed with, I'm not a canon lawyer but... which instantly removes credibility from the answer.