I don’t think he’s untruthful as much as exaggerating the story (which the “we” usage also indicates). EquuSearch searched the area along with CID, FBI, Texas Rangers, and the Sheriff’s Office. The narrative being painted is that EquuSearch found everything and basically did the police’s work for them; that does not appear to be the case.
CID was the agency with primary jurisdiction, since it concerned an Army soldier missing from an Army base. FBI was present on the site when the body was found. It doesn’t appear any investigative bodies were excluded from relevant information, and the timeline presented in the FBI’s affidavit seems to reflect this. One could even argue the main reason their main suspect fled was exactly because EquuSearch and others were releasing way too much information to the public, to the detriment of the case.