After hours of searching... it seems theThe ESRIJSON driver that is responsible for FeatureServer does not support handling multiple layers at the same time, you have to query layer per layer.
For example:
ogr2ogr -skipfailures -f GPKG floridaData.gpkg "https://services.arcgis.com/V6ZHFr6zdgNZuVG0/ArcGIS/rest/services/COVID19Surge_Florida_InterventionScenarios/FeatureServer/0/query?where=OBJECTID+%3C1000&geometryType=esriGeometryEnvelope&spatialRel=esriSpatialRelIntersects&outSR=4362&returnGeometry=true&sqlFormat=standard&f=json&maxRecordCountFactor=30&outFields=*&returnExceededLimitFeatures=false&resultType=tile" -nln 0
ogr2ogr -skipfailures -update floridaData.gpkg "https://services.arcgis.com/V6ZHFr6zdgNZuVG0/ArcGIS/rest/services/COVID19Surge_Florida_InterventionScenarios/FeatureServer/1/query?where=OBJECTID+%3C1000&geometryType=esriGeometryEnvelope&spatialRel=esriSpatialRelIntersects&outSR=4362&returnGeometry=true&sqlFormat=standard&f=json&maxRecordCountFactor=30&outFields=*&returnExceededLimitFeatures=false&resultType=tile" -nln 1