No way Russian companies were recruiting and training a “private” army in Russia without Putin’s approval.
No way they would attack American allies (and some American Military personnel) in a battalion-sized combined arms assault with tanks, artillery and Engineer bridging operations, without Putin giving the go ahead. They crossed the deconfliction line established between the Russian and American militaries to conduct that attack.
Wagner Group could not exist inside the Syrian Theater of Operations without Russian Military authorization.
Russian jets, under the cover of the Russian flag, were supporting this offensive, until driven off by US jets with warning, close fly bys and flare/chaff discharges.
The air cover incident you mention was during another earlier incident not this one - the Russian-Iranian force had no air cover this time.
It’s nuts that this opposing faction of pro-regime fighters went right into the gates of hell and got blown away. They must have had some idea what they were getting into. At the least, it looks very unprofessional and could be the result of bad leadership, who knows. It’s like Mattis said, they were perplexed as to why they were doing this, laying portable bridges over the Euphrates, firing at the SDF troops.
FYI: The Wagner Group operates out of Hong Kong, not in Russia where such operations are illegal. It recruits Russian veterans. It may be under direct Russian supervision, but that is not confirmed and could be independent - there are several schools of thought here.