The proof is in the over 200 million people that constantly buy consoles and handhelds gen in, and gen out, with the exception of current gen handhleds which lacked the most important feature of any mobile device (Mobile Data and Communication). There are over 200 million consoles in homes, and some of them are used by multiple gamers, which means that the poularity of a great handheld should outstrip the popularity of a great console. This has been prooven gen in, and gen out buy Nintendo fans who buy 4 to 5 times as many mobile devices as home consoles.

A phone built on a scaled back version of the PS4 hardware on 10nm FinFET would be no where close to $1000. Industry leading smartphones from Apple and Samsung only cost ~$250 to make, these companies absolutly rape consumers with the $700+ price tags they put on them. In the handheld and console industry things have always been about taking a loss or breaking even on hardware and making profits through software. PS4 cost ~$380 to build at launch and sold for $400, PS3 cost ~$800 to build at launch and sold for $500. 

If Sony released the above 10nm FinFET PS4 Mobile device, that could run all PS4 software at 540p, and checkerboarded to 1080P, and had 256GB of on board storage, the device would retail for no more than $400. The build cost would be somwhere in the high $300 range. 

As far as apps went, it would be identical apps for both, because it would be the same hardware. Why would they make apps for a console? Because, consoles are the future of home computing. It is the reason Scoprio will have every banking app, every torrent app, every video player, and on, and on, and on via Universal Windows Apps. Consoles are to big of investments, and far to capable for these companies to let them sit around for games. These devices are multiple times more powerful then any other device in 99% of homes, and they trying to use them to unify, and elevate the lowest common denominator.

On Nintendo having to change their business model. That is absolutly correct. Yes they would have to, and they absolutly need to. The console and handheld business model of releasing stagnate hardware for 5 to 8 years is no longer viable. When you only had a handful of powerful companies releasing new hardware every couple of year that was fine.Now when you have dozens of companies releasing revised hardware every year, and entering the same market space, you have to evolve or die. Sony and Micorosft are doing it right with Scorpio and PS4 Pro by starting before they get overtaken. Nintendo and Sony did it wrong with Vita and 3DS, and now we are left where we are. They have to change or they will loose the mobile market, and if they loose mobile, the home side will eventually colapse to these more hungry competitors.