Looking for low-end uC families with LCD controllers

I wouldn't think a LCD segment knew or cared whether it was muxed or static driven as long as it's vsat, voff, freq, and dc bias req. were met

Users would care if the supposedly off segments are ghosted (dimmly lighted).

But we are talking about driving it beyond spec (3.0V +/- 10%). 2V to

6V LCDs are unusual, I don't believe it until I actually tested them. To compensate for wide battery range, you have to over-drive the LCD most of the time. For the 2V to 6V LCD, how many units have you tested over how many years? How much do they cost? Our 3V LCDs are $0.60 each for 10K Qty.

Our factory (ISO 6 clean room) makes thousands every day, but they cannot make them (cheaply) over 3.3V. I think it would need very pure glass for good isolations. Perhaps you can build a class 1 (ISO 3) cleanroom for it, but the LCD would cost way more than $1 each.

Which has what effect on their lifetime ?

What is their spec - surely wider than +/- 10% ? - is that 1:1 Mux, or higher ?

I would have expected the duty cycle to matter : so a 1:1 MUX LCD will be very tolerant of voltage, as it is driven well past the optical switch point, in both cases. with 2:1 3:1 etc, then you are relying on the partial voltages that exist in OFF conditions, not being close enough to the thresholds to cause noticable ghosting. That would also mean the viewing angles of MUX mode displays would be less.

We have a 4:1 (1/4 cycle) 12 segments x 4 commons LCD. From testings,

10% is noticeable, 20% is unusable.

Overdriving 1:1 (static) would not be noticeable, but shortening the LCD life.

You should consider the Renesas H8/300L Super Low Power series for low-cost LCD MCU with many variants of chip sizes with differen combinations of SEG and COM pins.

A chip like the H8/38004 is cheap, and you can get a development boar from many distributors or even Digikey.

Did you Try Keil's Chip finder for the 8052?