Car share has probably fallen out of favour thanks to the pandemic and I suspect it will take some time to recover.
Longer-term though, car share is more flexible than public transport in terms of the route taken (it will generally be door-to-door) but less so re timing. It needs two or more people who can work or have to work pretty much identical hours which is if anything becoming less common as working practices change. Leisure trips (going to the same show) are more likely but that probably happens more generally already.
I did car-share with a colleague in 2000 - his car had some fuel in it, mine was almost empty at kick-off time. Even though we lived fairly close, and worked the same hours most days, we found it hard to cooperate closely enough and did not persist with the arrangement.
Since then, hours had already become more adaptable, and now the common days of travel to work will be fewer for office workers at least. So presumably the scope for this kind of car-sharing will be much less too.
It would take a big "push" from outside factors to make up for this lack of "pull" factors (i.e. those relating to the people concerned). Maybe something like the big green fuel price hike implied by the government's decarbonisation plans, dressed up in some way that placates the Tory back benches (and others) but would not help office workers commuting by car?
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