What are the advantages of a lapel "coat stick" pin over other pin styles?
The ones you call coat stick pins have a much longer pin and will therefor stay upright much better than the ones with a shorter pin.

The top kind, with just the short pin and the butterfly clasp will turn around and you will have to position the items on the front in such a way that it can be seen in any position and is likely not suitable for the kind of work you mention.

The second kind, the safety pin with or without a locking mechanism, will stay in the position you pin it on. But with only the short length, it is likely to lean away from the fabric you pin it on. This is more a problem with light fabrics and loose clothing, like women often wear, less with mans suits.

As I started out, the third kind, with its longer pin has a much more stable ground for the pin and will allow for less movement of the finished product.
But still consider the lighter fabrics and possible movement of the garments.  

For boutonnieres to be worn on a suit coat, I would personally prefer the last kind.
For a boutonniere to be worn on a shirt, blouse or dress, I would prefer the second kind, the locked safety pin kind, and design the arrangement such that the weight of the flower drops down from the pin, not being above the pin at all.