I've often wondered about what role Baba plays, exactly. And if he's just PR, then who's really pulling the strings? Who let Xillia get released unfinished?
He's a producer, which doesn't mean much as depending on country/company/project it could be almost anything.
He's clearly not someone directly involved in development though (to make it simple he's neither a programmer or a graphist thus doesn't oversee any of these aspects by himself), it clearly showed when he's away for a whole month (his usual summer trip) when the dev team is most likely in crunch time at the end of their project (Xillia, Xillia 2).
Overall he seems to have taken the job which was mostly done by Yoshizumi alone before he came up in 2006-7, which was to be a middle person between Bandai-Namco and the Tales team before it merged fully with Bamco in 2011.
That would explain why Yoshizumi left in 2013-4 (his job was made redundant with Baba and the merge).
As for what he does it is, most likely, general management and planning for the dev team including ressources and budget with what Bamco gives to the Tales brand.
Regarding marketing and strategic decisions (eg. which platform to release the next game on, when that would be...) it's most likely decided by a board in which he's involved, and when going into further details for a project (what gimmicks will define the game, what is the theme, the name, what will be the overall plot etc.) he should be the one leading the decision making with the staff involved.
Him not being fired after Zestiria necessary means that he wasn't, at least, the only person involved in that fiasco imo.
And about the overall quality of latest projects I'd say it's a mix of mismanagement, tight schedule, too much ambition and not enough budget and skill to make up for all of that.
Also sales didn't suffer that much, so if they can get away with it they don't have many reasons to improve their dev process.
