Right, it depends on the company and the roles. Sometimes they're there to functionally facilitate the project, sometimes they're there to creatively steer it.
In big Japanese companies, I would say the producer is critical in maintaining the brand representation from beginning to end of the project: if you like games "by Sony Japan", you like the work of the producers.
Look at Mr. Toriyama's resume: Bloodborne and Demons Souls by developers From, Soul Sacrifice developed by Comcept, Folklore by Game Republic, Monster Kingdom by GAIA, Freedom Wars by DIMPS, Echoshift by Artoon... none of them are developed in-house by SIE Japan. Sony in Japan doesn't really have a lot of internal productions (Puppeteer, Knack, the Ico games, they own Polyphony Digital; some other stuff not coming to mind, but not much in recent times.) How SIE Japan works is that the producers
produce the games for the company's imprint. They find the talent, take pitches and/or pitch their internal ideas, set budgets, rubberstamp creative decisions, solve problems, coordinate team efforts across the teams at all companies involved, plot the marketing approach, etc. Sometimes they're vital to the creative process, from idea to execution; sometimes there's a strong creative vision brought to the project already by the director and the producer is there to give him the tools to realize it.
This approach BTW is not unique to Sony, of course. It's prevalent in Japan, and has been since the early days of gaming. (Dragon Quest is Enix's claim to fame and is Yuji Hori's baby, but even the very first game was made externally by Chunsoft under "Director" Koichi Nakamura, and up through IX it was teams at Chunsoft and Heartbeat and Level-5 making the games under Hori's Designer/Scenario stewardship.) It's also not just a Japanese phenomenon. In his Incognito days, David Jaffe was somebody who kind of "middled" between producer and designer in the early Twisted Metal days by working logistics as well as design at the home base at Sony in California while collaborating with the SingleTrac team in Utah. Every Ubisoft game now is a collaboration between a head studio in one territory and a number of satellite studios, I assume with a lead producer at Ubisoft France and/or Ubi USA making sure every Assassin's Creed follows the playbook. (That was Jade Raymond's role for the first two, but I'm not sure who's the main person for AC these days?) It's also not exclusive to games. Jerry Bruckheimer has AFAIK never touched a camera, but if you ever go see a movie produced by Jerry Bruckheimer, you f***ing know it because he has a kind of "house style" that the directors he chooses follow.
Shigeru Miyamoto himself is known as the greatest game creator, but
look at his credits and you'll see relatively little Director or even "Designer" roles; he's generally working as a Producer, but a Producer whose heart can be found at the center of all the projects he oversees.
The Producers rarely get famous in gaming, (heck, you're lucky if a regular game buyers even know the development company on their favorites, much less the creative leads,) and the fact that their role cannot be as easily described as a "Director" makes it hard to tell what their importance is on their own credits. (I think the reaction of "Sony Japan is doomed!" is a bit of hyperbole... although I do worry about SIE Japan and it's fading presence since basically the Vita died.) But Producers can be vital to a project (and IMO are more often integral to great games than in other territories,) and it'll be interesting to see where Mr. Toriyama ends up. Sony has a weird habit of signing projects with staff who have left the internal studio (Fumito Ueda and Eat Sleep Play, for example) so maybe there'll still be ties, or maybe not.