He was executive producer though, if there was something he didn't like, he would have thrown it out. Developers just make the game, if the controls were unresponsive or it looked like crap we can blame the developers. Unfortunately everything was on point and the only crappy thing about XIII would come down to an executive decision on where the direction of the game was going. Decisions like taking out the cities and shops to streamline the story, ect. It is exactly the person in Sakaguchi's chair that screwed up XIII.

Not EVERYTHING is solely the responsibility of the producer and if anything it's the DIRECTOR that get's to make those type of decisions. The producer is only there to manage the funding, contracts, and legal issues etc ... 

Bloodborne is going to have one example where its possible, its sony japan and fromsoftware. Two of the best studios in japan

The Director is responsible too. But saying as a producer he couldn't (or wouldn't) have any influence of the direction of the game is wrong.

Has SE not made great games themselves too after the departure of Sakaguchi ? FFXIII is more approved by critics AND has also sold more than any of those games you mentioned but not only that they've made some great games for handhelds too so doesn't that prove anything of SE's worth ? Money sure can be a problem but you can't deny the chances he had for his studio to grow and if he truly did have skills Microsoft or Nintendo would've swooped him including his studio up by now since the title "creator of Final fantasy" isn't something minor ... 

I say yes FFXIII would've still stayed the same in part due to the fact that he hardly ever handles the development directly anymore. As for the following there's no guarantee that those games you mentioned would've done better had it been published under SE's name. The name means little when it comes to making a game succeed, what matters most is the quality of the game and the talent that the developers had ... 

You're greatly underestimating the role of a producer. Miyamoto was a producer for Metroid Prime and he's the one that handed Retro Studios the IP based on an Action Adventure game they were developing (which he forced them to cancel). The game was originally going to be in third person, but Miyamoto scrapped the idea and forced them to adopt a first person perspective (which the development team wasn't too happy about at first).

I didn't say that ...

They may influence funding of a certain aspect of the development but that does not mean that their influence is the greatest of them all when it comes to developing a game ... 

The direction of a game comes solely down to it's directors and developers ...

Considering the games they released after Sakaguchi left... they can't have been the good ones