It's not running from your computer, that's the whole point. I mean I guess that is sorta iffy since it is still effecting a consumer directly, but the consumer doesn't physically own the server... so... I dunno.
Let's not involve physical ownership here, though, let's try to stick to use cases.
Because consoles are literally just specialized PCs. Mobile devices technically are too, but to a much greater extent. It's sorta like divergent evolution, with consoles being more closely related to PCs.
What if next-gen consoles ended up being closer related to mobiles?
It should also be noted, though maybe not entirely relevant, that many consoles have been directly referred to as computers by the companies that created them (Famicom being the "Family Computer", PlayStation Computer Entertainment System, etc.).
All those names date back to a time when the word 'computer' was still hip. How many 'computer entertainment systems' do you know developed in the past decade?
Why are you asking these questions exactly?
Well, I'm trying to figure out what makes you think x86 is so 'modern' in consumer aspects.
If our cloud-providing server farms soon run on POWER and/or AArm64, our cars - on embedded PowerPC and/or ARM (they currently do, and will continue to), and our smartphones - entirely on ARM, what is so darn special about x86 that you believe dictates the 'moder' characteristics of the architecture in consumer space? That people run their desktop browsers on x86? Or that two of the tree console vendors this gen went with Jaguars?