Is the problem that some people don't drive, or that so many people (including town planers) work on the assumption that everyone should drive and have access to a car?
Yes, it's a pain in the arse not to be able to drive, but it needn't be if there were fewer out of town shopping centres, NHS 'hubs' instead of local treatment centres, cinemas etc on high streets instead of on retail parks and so on. Town centres are jammed with cars (with attendant poor air quality) as people insist on bringing them into town for every visit, but houses are now being built on every blade of grass on the far outskirts, so there's not much choice for the elderly, infirm or those with young children, never mind the ones who drive 200 yards to go to the gym. We need to rethink the way things are set up, so there is a genuine choice about whether to drive or not. More public transport and fewer out of town centres would be a start. High streets are losing facilities, and even city centres (which tend to have decent transport links) are in decline, and more and more houses are build out of town on estates with no shops, surgeries or other infrastructure.
IMO there is a huge difference between accepting a lift to somewhere the driver is attending anyway, and asking someone to make a separate journey to take you somewhere. I can't imagine not offering a lift to someone who is going where I'm going (particularly if s/he couldn't get there easily otherwise) but would expect them to be ready at the arranged time and to leave when I am ready to go home, just as I would if getting a lift from someone else.