I said the Greeks began to establish colonies in Italy in the 7th century, not that the cultural transfer to the Romans began in earnest then.
But, from 6th century Rome, we DO have archeological remains from Greeks, for example temples, if I recall correctly.
Now, some of Rome's kings were definitely Etruscans, for example the Tarquins.
Thus, more narrowly, the identity-building project for the Romans would begin under the late kings and early Republic, say 550-450 BC.
This was, in its latter phase, also coincident with fighting the Etruscans, the previous overlords.
In order to prettify their own homely culture, wouldn't it make sense to borrow what they liked from an advanced culture like the Greeks, who were in cultural terms on par with the Etruscans, but not at all that menacing to the Romans?