A Periodic Review of PubMed for COVID Fecal Microbiome finally found some studies:
One paper reported very very good results!
The optimal eight oral microbial markers (seven faecal microbial markers) were selected by fivefold cross-validationAlterations in the human oral and gut microbiomes and lipidomics in COVID-19 
on a random forest model, and the classifier based on the optimal microbial markers was constructed and achieved an area under the curve (AUC) of 98.06% (99.74% in the faecal microbiome).”
“The heatmap showed that the faecal microbial community in CPRs (Confirmed Patients Recovered) was different from that in CPs (Confirmed Patients) and HCs,(Healthy Controls)” [SP is suspected Patient, SPR is suspected Patient recovered] which appears to confirm my hypothesis that most infections will leave a “garbage state” in the microbiome which will generally shift slowly back to the healthy state. This return to a healthy state is not certain and when it fails to happen, then we have diagnosis such as long haul covid, chronic fatigue syndrome, and post-infection syndrome.