From the Medicinnet article:
"Chigger bites themselves do not produce any long-term complications. However, because of the intense itching, prolonged scratching may lead to skin wounds that may become infected by bacteria. Chiggers in North America are not known to carry any diseases, unlike some other arthropods."
So it seems scratching is the problem, not the bites unlike mosquitoes.
Bite remedies:
"Many home remedies for chigger bites are based upon the incorrect belief that chiggers burrow into and remain in the skin. Nail polish, alcohol, and bleach have been applied to the bites to attempt to get rid of the chiggers by "suffocating" or killing the chiggers. However, because the chiggers are not present in the skin, these methods are not effective.
Here's how to get rid of chigger bites. Home remedies to help relieve the itching associated with chigger bites may help some people, such as:
Taking a cool shower or applying cool compresses
Sitting in a cool bath
Using bath products that contain colloidal oatmeal
Using certain over-the-counter (OTC) medications to help "
So, don't scratch and cool showers or baths to relieve the symptoms.