I previously worked for a spice manufacturing company. it is common to not have too many analytical tests. This is for a number of reasons:
1. unlike lots of other products, the chemical makeup is usually pretty standard as spices come directly from an organic plant source.
2. spices (if processed properly) tend to have a moisture content too low to harbor microbes and they usually are not an attractant to pests.
3. the spice is used for aromatic and flavor properties so testing for physical parameters such as color (other than visual inspection) is not necessary.
As a good QA manager, due diligence is important in the food industry and it's great that you are inquisitive.
While I mentioned the moisture content is supposed to be low enough to prevent microbial growth, you can still ensure this is maintained by (1) conducting environment humidity tests and (2) conducting random moisture-content tests and recording the data (you can send to a lab).
Allergens are a good topic to look in to as spices have the propensity to cause dust pollution or contaminate other products or other spices in the environment.
You can go the extra mile and ensure the hygiene of the equipment you are using to process your spices are up to code as far as food safety and hygiene are involved.
Hoping this helps.