The Director system did use the phone dial to map exchange names to numbers. The current keypad mapping is mostly the same. O and Q have moved, Z has been added.
Indeed - but the current push button arrangement did not exist on GPO telephones in the same time period as
published alphanumeric Director codes: after experimental phones in the Director era had two rows of numbers with letters, the modern push button arrangement appeared
without letters on the GPO 758 telephone in 1976, because the UK had converted to 'all-figure dialling' from 1966. Even where the numbers were still based on the original letters (they usually were and often still are), the letter combinations were no longer published and there were no letters on UK dials or push buttons for about 30 years. There was therefore no need for 1 to be top left to have the alphabet starting at the top of the keypad: with no letters on the dial, any logical numeric arrangement could have been used.
The separate taps used to be for good reason. Either you had non potable cold water along with low pressure hot or mains pressure with low pressure hot. Obviously mixer taps definitely don’t work with the latter. Nowadays with combi boilers you have both hot and cold at mains pressure.
And yet mixer taps in kitchens have been common for decades, happily 'mixing' mains cold with either low or high pressure hot. As long as the water only 'mixed' mains cold at the open outlet of the spout, there has been nothing to prevent mixer taps in any room (our 1970s bathroom had a mixer on the basin, our 1930s bath had one too); and what are showers but extended mixer taps? The two-taps thing is much more complex than it appears.
I would argue that the single-lever mixer tap is not very good design: with the 'default' central position usually being a mix of hot and cold water, there is a tendency to draw off both hot and cold whether or not it is really needed, and whether or not the water actually runs hot by the time the tap is turned off, wasting hot water.