I agree, a chip specifically for driving LC display glass would make life so much easier and they are quite common, with many designed for 7-segment/numeric displays. You would of course have to drive that with a microcontroller, but it should be as, if not more straightforward than the character HD44780 drivers.
As for the connector, they are made to be independent of 'pitch', so as long as you get the pitch of the PCB trace connectors right, the alignment of that connector is non-critical, it simply provides a soft surface so pressure can be used to ensure a good connection for signals. For prototyping using an LCD from a old nokia phone (5510, I think...) I cut lines in blank single sided PCB material, and just soldered wires to that, with the LCD and rubber connector clamped on top - works a treat for developing software!