When I intitially raised my first batch of them I exclusively fed firebrats, but once they reached adulthood I started feeding various other things. I pumped females full of bottleflies, locusts, waxworms, or mealworms and never noticed anything unusual. No signs vomiting, weird poop or problems with ooths. Young nymphs of the next generation were fed springtails at first and for later instars I also gave them a bigger variety and never noticed any complications.
Eremiaphila is a pretty large and widespread genus, perhaps species that originate from extremely hot regions do have issues with such diets while others don't. Many of the imports that made it into the hobby never got their species identified and the location of origin often ends up forgotten, maybe that is also a cause for mistakes in their care.
I personally raised E. rotundipennis, which originated from Gujarat in India, and they felt like pretty uncomplicated mantids to me. Fast growers, ravenous eaters, and very productive females.