You overlook a crucial aspect when you say that Suarez should move/run more. He can't do that, not for an extended season anyway. These issues haven't appeared last season under Valverde. There's 2 seasons now that Suarez is slow and his off the ball movement is poor. Every other run of the mill defender can match him for pace and even exceed it. As far as I see, he doesn't even move that much more for Uruguay, fact that strengthens my point. Considering that his link-up play and first touch is also poor, what does he offer to put us a level above? Please take into consideration that the guy is treated as the second best player of our team with wages to match.
He offers all of the above in smaller doses. That doesn't mean he should be replaced. Look at him at Barca vs him for Uruguay. When he knows he has a manager he intensely respects and the teammates he can trust, he has no problem putting a shift in. He couldn't do that at Barca a lot of the time when he knew half of the team doesn't like the formation anyway.
You are talking like Valverde is a proven winner and these shit indoctrinated(with tiki-taka i take it) players are the ones at fault. Yeah, no, he was the one telling the players to defend in Rome and the one who reacted by bringing in fecking Gomes of all players to defend some more at 2-0. Sorry, but that man has another thing coming if he thinks those tactics are suited at Barcelona and should not be followed blindly, unless he realizes that it's not goddamn Bilbao that he manages now. He has to prove it this season to be respected ala Luis Enrique. When the later came in we changed from a heavy midfield play to a more counter attacking style with the focus on our 3 forwards. Seems like the players were not ingrained with some dogma then. He also benched Pique when he underperformed.
Players like Pique playing through an absolute lack of form is on the manager. Valverde is the one acting cowardly by not benching Suarez, Pique, Busquets etc. at various points. Like an expression in my country says, the fish begins to stink from the head.
He isn't proven and they ARE indoctrinated. Every long-serving player is.
Food for thought way of saying this though...
Consider some facts:
1) 5 Barcelona seniors (Pique, Busquets, Alba, Iniesta and Messi; 6 including Masche who was out the door) have never played together in anything other than a variation of a 4-3-3 at Barcelona (sporadically in a 3-4-3 maybe).
2) They have equally never played a season without Alves before 2016 who is an all-time-great #8 playing from RB.
3) Rakitić is the only utility player to have played in a 4-3-3 with Xavi leading it (and someone can correct me if I don't remember this right, but he's rotated in for him more often than not? him or Iniesta).
4) The last backup forward Barcelona seniors treated with minimal trust was.. who..? Pedro?
So this is not only the first season you have had with a different formation, but also with all the utility guys actually having bigger shoes to step into in this new Barcelona. And first season is always a prototype for these things, you're meant to see how it looks and iron out the issues.
Cowardice isn't really the same as trying to have healthy respect to veteran players you assume would be professional and understanding. Change of formation was enough of a shock to their systems. I believe he was soft on them, but he tried to be careful. Being rebellious and coming in with muddy shoes onto shiny Barcelona carpet and changing a philosophy to address an obvious issue with direct benching of senior players who do not conform to the new order is not always the way to go. Not only did he have to choose his incarnation to combat the identity loss, but also kind of fight the board's opinion on it to get players who would ALSO have to fit in between the seniors, who were used to playing with self-sufficient leaders like Alves, Xavi, Masche...
I can kind of understand why he seems to have protected Dembele from the toxicity that comes from fighting through the doctrine. The guy has YEARS of service for the club ahead of him and his inclusion has to go immensely slow. Otherwise those greats will simply turn around and say "aaah, this kid will never get it, it's too difficult". The Suarez and Messi attitude and lack of patience for Semedo at times knowing he's an outsider... Imagine how the personalities in that dressing room have changed over time. 5 years ago you would have had the corner of 4 or 5 tactical geniuses discussing between each other what they ought to do if things aren't going right. Now? It's the Pique-Alba-Suarez corner with outsiders looking in.
I think many of Valverde's decisions are well grounded though on the face of it. Costly in terms of rotations, but this is a VERY difficult dressing room to manage. And I don't think there's a manager out there who could do a better job right now.
I think Firmino could be great (though he'll be 28 next year so not exactly a very longterm solution either) but good luck negotiating with Liverpool again after the whole Coutinho saga. 
