Correct. There would be no death certificate in Michigan or anywhere else. When an individual died while on active duty, a standard form, Report of Death (WD AGO Form 52-1), was executed. This was used by next of kin for any purpose that normally required a death certificate. If death occurred in the Continental U. S., military authorities could - entirely at their descretion - provide the local coroner with a copy but this rarely occurred. The next of kin could likewise file a copy with their county clerk but this was also a rare occurrence.
You might want to order a copy of his deceased file. You'll find practical instructions on how to do so at message #2112 (go back to the main page, use the box in the upper right corner labeled "Jump to #_____", then click on "Go").
The file will contain a copy of the Report of Death and, possibly, the exact location and circumstances of his death.