I remember seeing this all over the place the last couple of years before the PS4 & XBO were released. So many of the arguments were based on an actual observed fact: a decline in console game revenues during the early 2010s. It seems they were extrapolating from that to argue that the console market was hurting, and that perhaps the emerging mobile gaming segment and other "disruptive" developments were the reason. What they failed to notice is that the console market was cyclical, rising and falling like the tides. Every single Gen 7 platform was in decline by 2012. I honestly got kind of tired of having to point that out so much back then.
Now, it is true that the Vita and Wii U were commercial flops and the 3DS failed to replicate the success of the DS (though doing that well is a hard task, to be fair), and I suppose that might've been worrying signs of the market's health to some. However, Sony & MS were doing fine on the home console front. Nearly all the variability in the market was with handhelds and Nintendo's home consoles. Conventional consoles (i.e., PlayStation & Xbox) were and still are doing just fine. While there was a dip in combined PS+Xbox sales in 2012 & 2013, they rebounded back in 2014. Combined PS4 & XBO sales were on par with combined 360+PS3 sales, and, despite stock issues, the PS5 & XBS are doing well.
I always though the idea that emerging "disruptive" forces would've had an effect on the home console market. PlayStation & Xbox offer experiences that you just don't get on mobile. Just look at what kind of games dominate on consoles. Elden Ring, God of War: Ragnarok, and Horizon Forbidden West were among the Top 10 best-selling games of the year. You just don't get that sort of thing on mobile. What you do get on mobile is stuff like Candy Crush (still mega-popular over a decade later), Honor of Kings, Roblox, and Coin Master. Everything from the interface (sans expensive accessories) to the average phone's power to when and where people use their phones matters when it comes to the kind of experiences that people demand from that platform. Having a big, beefy piece of hardware plugged into your TV just never stopped being relevant, and I doubt it will any time soon. The idea that mobile is competing with home consoles is like the idea that potato chips are competing with steak dinners (I suck with analogies, so just roll with it).
One could have reasonably argued that mobile was eating into handheld sales, but even then, the 3DS was still offering experiences that weren't available on mobile at the time and it still sold well nearly 76 million units, and then in 2017 the Switch was released and utterly disproved the idea that mobile was some sort of death-knell for console gaming on the go. Also, it is worth pointing out that in Gen 7 the DS shattered all expectations, selling nearly double what the GBA did worldwide and nearly matching the PS2, and the PSP was the first (and only) time a non-Nintendo handheld was a huge success. So, the handheld console market was arguably over-performing back then. By the end of 2010, the DS & PSP represented about half of all handhelds ever sold at that point. They sold ~234M combined in a single generation compared to 200M for what the GameBoy & GBA sold over a span of nearly twenty years. The 3DS was therefore more of a return to normal (though it still arguably underperformed thanks to a slow start). The Vita, meanwhile, failed for reasons that likely had little to nothing to do mobile (its high cost—exacerbated by expensive proprietary memory cards that were sold separately—combined with lack of software support are likely the main factors.
TL;DR: The early 2010s claims that consoles were in danger was always based on a very superficial understanding of the market, one based largely on a short-term decline in revenues.