It's a little more complex than that, IMO.

For starters, IBM did also do the CPU for the GameCube, and that was likely a part of the Wii using an upgraded PPC arch for that.

On the flipside, there is the theory (I think even Copetti brings it up in their XBox 360 Architecture breakdown [0]) that IBM using the Cell PPE for the 360's tri-cores left a sour taste in Toshiba, but more-so Sony's mouths.

I think the big 'X factor' though, was that, for as much pain as it caused AMD in the short term, (it's so easy to forget their 'malaise' era, i.e. Early Bulldozer and the GloFo split pains to their margins[1]), AMD made all the 'right' choices to let the console vendors have their cake and eat it too.

Namely, AMD was more than happy to do a custom core if there was a volume contract (similar to what IBM was willing to for the 360/GC/Wii) but also now had a capable, in-house GPU. (And thankfully had Bobcat as a stepping point towards Jaguar[2])

There's part of me that asks, if AMD had an ARM core, if we would have all Consoles powered by AMD chips now. Nintendo likely bought into Tegra because it was an ARM core, and for better or worse their mobile stuff by the time of making that choice had 15+ years of proven ARM success (GBA, DS, 3DS) behind it.

(See also, Intel in the 2010s scrambling with half-assed promises of integrating custom functionality or FPGAs with x86 cores.)

[0] - https://www.copetti.org/writings/consoles/xbox-360/

[1] - From what I recollect, the GloFo split and how the contracts were drawn up as far as their production, had a huge impact on their ability to produce due to yields and thermals, as well as the contracts for how GloFo got paid; it was at least part of them diversifying with TSMC as soon as they reasonably could.

[2] - As a Rant, I am pretty sure, if Jaguar had Desktop/Mobile versions that included a Dual channel DDR controller, they would have cleaned up on the low cost laptop market. I had one with, I think it was an A5000 or A5200, and for how tiny the battery was it could last wayyyy longer than any of the intel laptops I had for the time, but churned if you were doing memory heavy stuff.

Having a presence in the consumer market is not a requirement for being innovative.

IBM stopped building servers based on x86 because the margins were too thin for their tastes, but they never stopped building on top of POWER and Z.

Xbox 360 used an Intel CPU.

PS3 used the Cell processor but it’s debatable how much that was an asset vs handicap. Total PlayStation sales dropped for that generation vs both PS2 and PS4. The manufacturing costs didn’t fall nearly as fast as expected and it was a poor fit in terms of cross platform development etc.

PS4 moved to AMD.