You can have an opinion whilst still backing him as a supporter.
The club makes the decision when it's time for any manager and/or player to go, the fans don't. Us not supporting them, be it a player or manager, does nothing but make the situation worse.
Again that doesn't mean you can't have an opinion, and from what I can see that's fine on here as long as you out your point across well, but to specifically not back him goes against what the number one thing of being a supporter is IMO.
One of the reasons people get pissed off at fans turning on the manager is the mood of said fans does tend to get taken into account when owners make decisions on these matters.
If said fans feel like the manager is doing a poor job with the resources he's had its fair to say they will express that because they feel making it clear that the status quo is not good enough is important.
Most would agree that Rodgers has done a poor job this season and over the summer. He's basically thrown away a treasure chest and written off a season with Inept performances, poor training, naive tactics and baffling team selections.
We are not a Chelsea or a city that can just throw money away. When we buy a 20m player (or two or three) and they aren't good enough it sets us back a lot...if we are to climb the mountain it is with baby steps, making the right decisions over and over and building to a crescendo. If a manager cannot make correct decisions in the market then he is not right for us.
My doubts over Rodgers are not exclusive to this season really. I look over his entire tenear here and I am not sure he is capable of building a team to compete. Last season cannot be discounted, but nor is it the entire story.
For my money he deserves at least January, and probably until the end of the season if it at least stays respectable.. But I can't judge people who have decided one way or the other....none of us really care whether Brendan Rodgers is successful, we all just want Liverpool to be.