A joystick isn't technically a button. No one's asking for the B/A/X/Y/dual analog stuff ... that will never fly on a standard smartphone. People don't want all that.

From the company's point of view, Why make a smartphone catered to gamers when you can just make a smartphone catered to everyone else in the world?

Casuals still play games on a non-gaming smartphone...

Which is exactly why having just a joystick would be fine. That would make a ton of games much more playable for casuals and general gamers ... the controllers with the X/Y/B/A/dual analog/d-pad stuff get it wrong because no one wants all that, it's overkill and makes a phone look ridiculous. 

No one wants a starts a three course meal and wine on the menu when they go to a burger joint. But that doesn't mean there isn't room for things like apple pie/ice cream deserts and a fish sandwhich either. 

Adding a joystick to a smartphone would already look weird.. If you don't have a physical keyboard on your smartphone you can't place it anywhere without making it look super strange

Yeah, you could try attracting the gamers with this phone but it actually targets a very narrow range of consumers.. Gamers usually would just go for a handheld and casuals, a normal smartphone. Company's always want to profit so unless there is a demand for this, they will continue to make phones bigger and have a better camera :/

I hope the joystick wouldn't pop out easily. That become annoying when it snags on your pocket. But could apple have the finger print scanner work within a joystick.

The perfect gaming smartphone was already released, and it was called PlayStation Vita 3G/Wifi.

I feel terrible that it took me almost three years to realize it. The even bigger issue is that Sony does not even offer a 3G or 4G Vita anymore. So honestly, anyone that has the 3G Vita is amoung some of the luckiest people on the planet.

If Sony revamped the PS Vita with 4G and put the Xperia Z3 battery in it, then they would have hands down the best mobile device on the market. A handheld gaming device with 48 hour battery life, and a giant library of PSP, PSOne, and PS Vita titles, plus access to PS Now and Remote Play, what gamer would not love this device?

Currently I use my 3G Vita as a Phone via Skype. I only get about 4 hours battery life, but I have a charging dock at work, and at home so it is no issue to me. This is by far the best phone set up I have ever had. No android or iOS device on the market even scratches the surface of the experie the Vita offers, and the Vita OS is miles ahead of them for everything I need from work, to home and everything in between.

Question to those of you who have a PS Vita: What is it like controlling a game with your index finger on the backside of the controller?
Point is, would it be a viable option to put some sort of joystick on the backside?

I personally would not be able to use it. For me the touch screen and touch pad do a decent job as subs for R1/L1 and R3/L3. Otherwise, touch is just not accurate enough to replace physical. I have found anything more than a tap against the touch surfaces almost never works unless the action is accompanied by an on screen indicator, which imo defeats the purpose of a handheld, as loosing screen space to controls is one of the things I hate most about gaming on smartphones. 

Even if a joystick was placed on the backside, the ability to control it would be greatly diminished in comparison to joysticks controled by thumbs.

Tagged, interesting points. It's really baffling that up until now, and despite so many people looking for it, nobody has found a solution satisfying enough for these issues yet. This also means that the separation between phone casual gamers and dedicated HW ones will continue at least for a while, most probably at least for next gen too, as even if somebody devised a good solution now, developing a whole HW/SW/services platform requires a lot more time than designing that spanking new phone model each year, that despite new is a smooth and continuous evolution from past ones, with mandatory total BC on the main functions a phone must have, and deals with telcos for the most important new functions that may be introduced at major new designs.
But if there is no place for a gaming phone unless the additional HW interfaces are integrated in the design in the smoothest and seamless way, there should definitely be a place for portable consoles with phone capabilities, if Sony removed them is probably just because underwhelming sales didn't allow economies of scale sufficient to include the feature while keeping the price competitive, but with costs dropping and the user base slowly growing and realising such economies on the console's main components, maybe a version with phone capabilities could become viable again (initially it was, as the premium model had an acceptable price at launch, it became unacceptable due to total sales not growing fast enough after running out of early adopters).
Anyhow, time is running, the longer it takes to find a satisfying solution, the more viable another generation of dedicated portable consoles becomes, and it obviously applies also to the generation after the next, probably even Ninty will be cautious developing it, and it will push its project of an integrated platform, but if within two years from 10th gen launch, a gaming phone layout able to attract both casual and dedicated gamers won't have been devised, it will green light of the definitive realisation of another dedicated handheld, be it an autonomous platform or integrated with the home console one.
About Sony, who knows, it has good HW and it could have good ones in the next gens too, but it lacks focus on that market and its PR for it is appalling.