I'm not sure I agree with you here, at least with the example you have provided.
After the Chelsea game, Mourinho came out and flat out lied to the cameras in the most classless way you can. The media lap it up and the back pages react to it, but over the course of the next few days I saw and heard numerous pundits talking about how wrong Mourinho was regarding the penalty and Eto'o.
Rodgers, for me, handled the situation correctly. He had, 3 days earlier, laid into a refereeing performance and been put under pressure by the FA. For him to lay into Webb straight after Chelsea would have been risky to say the least. Instead Rodgers sarcastically spoke of what a great performance Webb put in. This was clearly a subtle attack on webb, without saying anything controversial.
The media will always lap up outbursts from Mourinho and I'm happy we don't have a manager who acts in this way and the only way Rodgers could have competed with that level of attention would have been to come down to his level. I don't want that.
The battle with the referees is not won after a match, it is beforehand. The best examples of influencing referees are when managers make statements in their pre-match conferences about a player or manager or act (like simulation). The likes of Wenger and Ferguson were masters at this.
I'd like to see more of that from Rodgers, but to be fair to him he has built a good relationship with the media and they do seem to take on board what he says.
You have to remember we are rebuilding a relationship with the media and FA that was all but destroyed a couple of years ago. You can't go ranting about anything without some support in those establishments, otherwise you just look lost.
Rodgers is doing fine. both he and the club could improve in their comms, but I'd rather we didn't start taking a leaf out of Jose Mourinho's book.