Assuming those evidences are true, then why must they mean a spread to the steppes essentially in the 5yh millennium BC and not before? There was CHG-like admixture even in Mesolithic EHG, I would not be totally surprised if a relevant part if the CHG affinities in the steppes (at least its southeastern portion) dated to even before the arrival of the Neolithic package there and increased even more after it. I think samples going from roughly 6000 to 4500 BC could help us a lot to determine what happened. By 4200 BC the steppes were already full of CHG and later there was IIRC only incremental increase of an admixture process that was already nearly complete by the Chalcolithic. That is why I think there is little more than wishful thinking in linking Maykop or any post-Leyla Tepe Caucasian culture to the steppe cultures that would participate in the origin of Yamnaya and CWC.
I don't see any CHG in Samara 5,600 BC, nor in Ukraine up to 4,700 BC. The first time I've noticed it appearing is in Khvalynsk 4,600 BC, and at very variable levels (between 0 and 27%) making it look like a recent addition to the mix.
CHG appears heavily in a Khvalynsk sample that best-fits with Eastern and Southern populations, but with North Caucasus more than Iranian, so I think a Western Caspian origin looks a good possibility.
My calculations suggest four separate lasting early CHG pulses into mainland Europe, each relatively minor, but with cumulative effect -
1. With the Neolithic expansion from Anatolia into North Central Europe - 6th millennium BC
2. With the migration of Suvorovo from the southern Steppe into the Balkans/Carpathians - 5th millennium BC
3. With a migration from the Western Caucasus (Maykop?) into the Balkans/Carpathians - 4th millennium BC
4. With the migration of Yamnayans into the Eastern Baltic - around 3,000 BC
(There were also pulses from Yamnaya further South, which I see as largely fizzling out)