I think the last completely privately owned company to be of any significance in Eurovsion was RTL (and I also think it was the only one).
Mainly because in countries that do have a commercial broadcaster as part of the EBU (like France, Norway, Sweden, Finnland and the UK) those broadcasters are just the 2nd EBU member besides the public broadcaster. And the public broadcaster usually do the Eurovision.
So the new Greek broadcaster will be the second (Assuming Privatization is a complete one and not some form of hybrid.)
And of course such a broadcaster may not be eligible for EBU membership depending on how it is all set out(AS EBU members have to have some form of PSB function).