The idioms are not being well used, since “first hand experience” and “cutting one’s teeth” have different origins and connotations. Taken as a whole, they form a mixed metaphor.
For example, first hand experience refers to actively doing something onself instead of being told about it second hand.
In contrast, cutting one’s teeth is essentially passive. Infants like to have “teething rings” and other objects to press their gums against, but these are not like blocks or other developmental toys. When you say that someone “cut his teeth” on something, it typically means that they experienced it in infancy. In some cases this can be interpreted as the infancy of a career, but it in the example you gave, it sounds odd.
The problem is that the apparent intent of the writer is to describe formative experiences that lead to an individual’s progress in a career. However, the grown-up metaphor “first hand experience” is followed by the infantile metaphor “cutting their teeth”. So the metaphor is not only mixed, but backwards too.