I wonder if the contrast pot is OK as the adjustment in quite limited i.e. very narrow, in the beginning of scale.
Absolutely. The contrast itself is the actual voltage between "Vo" - pin 3, and the +5V rail. The 2k2 resistor chain R1 to R5 divides this up into a set of steps which are used to multiplex the LCD display which is very sensitive to the voltage - that is actually how the multiplexing works, only segments which are strobed with a combination equal to the full
4.5 volts will actually show.
So I say 4.5 volts - this corresponds to about 0.5V on the Vo terminal, so the potentiometer is always set near to ground - setting it midway selects 2.5V which produces no contrast at all. In fact, a fair approximation to the correct voltage is a 10k resistor to Vcc and a 470 ohm to 1k resistor to ground. Just a 1k to 2k resistor to ground will probably work (since the internal divider already totals 11k). If fine control of the contrast was needed, the potentiometer would be a 1k in series with a 8k2 to Vcc, but in practice, this is nowhere near that critical and once set, it generally does not matter if the potentiometer is obscured by mounting the display (as happens with some versions of the "backpack").
Now a matter of interest - there is a 3.3V version of the display. You will note a mounting pad "U3" on the board and three jumpers of which "J1" is soldered as well as capacitors C1 and C2 (it is by the way, always a good idea to mount a 0.1µF capacitor across pins 1 and 2). On the 3.3V module, these components comprise a "charge pump" which actually generates a 5V supply from the 3.3V, a different jumper (or two) is soldered and the internal Vcc is now 5V.
Some variants of these displays ("extended temperature") require even more
voltage to drive the LCD, and so require an actual negative
supply for Vo which may or may not be supplied by the module.
And by the same token, you will note that if your 5V supply "sags" excessively, the LCD will begin to fade as a result.