And employ various aliens as their Front Spokespeople and their Muscle as well. Why bother putting themselves in danger when you can hire some Nausicaans to fight for you?
That's sort of how the Founders ran the Dominion.
Except in this case, it'd be different aliens each time to keep suspicions low. With the Jem'Hadar, they were there to specifically make it clear it was the Dominion.
Heck, why not have Ferengi Equivalents to the East Indian Company or the Hudson's Bay Company?