Edit: Looks like the Sony Z3C is a (and the only?) contender!
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But, I've also given up on that front. The last phone that was (barely) usable one-handed for me was the original Moto X. At this point, there is literally not a single phone on the market that (that I can find):
* Runs a recent version of Android (or iOS)
* Is <= 5" tall
* Has a recent/"fast" CPU/GPU
So, fuck it, time to give in and go with a phablet.
Sony Z3 Compact. Is 5.01 inches tall, but your original Moto X was 5.09 and you seemed to like it fine. Fast/recent specs. Runs a recent version of Android.
A height of less than 5 inches limits you to the Motorola Droid Mini, Moto E or the recently released LG Joy. (Those 3 run Lollipop and are from 2013 and later.)
Increasing height to 5.1 inches (well, 130mm), gsmarena.com shows almost a dozen phones that suit your wishes.
(Personally, I'm still not happy about having to give up on having a hardware QWERTY keyboards on my phone :/ )
4.7" is the screen diagonal. The total height, including the top and bottom border is listed as 5.44". This is according to Apple. http://www.apple.com/iphone-6/specs/
I think you and the parent are confusing the display size with phone size. The phone height is >5 inches. The display is 4.7 inches diagonally. The device size diagonally will be more than device height.
"Most"? I've not had a single Android phone where the buttons have been part of the display. A few tablets, but no phones. I'm sure there are plenty of phones that do that too, but I've somehow managed to avoid them for the 6-7 or so Android phones I've owned.
Pretty much every Android phone released in the last 2 years besides those made by Samsung (which is admittedly a large percentage) have had on screen keys.
I've had three Android phones of different brands released in the last two years, and none of them had on screen keys. Nor did any of the other ones I considered.
Edit: Looks like the Sony Z3C is a (and the only?) contender!
---
But, I've also given up on that front. The last phone that was (barely) usable one-handed for me was the original Moto X. At this point, there is literally not a single phone on the market that (that I can find):
* Runs a recent version of Android (or iOS)
* Is <= 5" tall
* Has a recent/"fast" CPU/GPU
So, fuck it, time to give in and go with a phablet.