Circulation may be down but if you believe newspaper proprietors no longer wield political influence then you really haven't been paying attention.
I believe that although there is some truth to this it has always been a convenient get out clause for the left's failure to be electable.
For example, the Sun likes to say 'It was the Sun wot done it', but there's a pretty strong counter argument that they're good at leaving it so late they know which way the wind is blowing and hitching their wagon to the winning team.
Also, social media has a far, far deeper penetration than the print press these days and is hugely dominated by left wing viewpoints. But they've totally failed to translate that into winning any actual elections.
I actually think in a lot of ways the right wing press is actually the outlet providing the 'balance' needed because of left wing success in suppressing opposing viewpoints in public life and large swathes of broadcasting media for example.