Cheaper: Questionable, unless the customer is unemployed. Tickets are practically always heavily subsidized, the taxpayers pay that bill. Maybe there is a city in the world where this isn't the case but I've never heard of it.
As far as convenience goes self-driving cars should be extremely convenient once we have them, no? I agree with you that public transport makes sense in very densely populated cities but most places in the world are actually wasting money on lots of public infrastructure that makes little rational sense, simply because it's a popular demand.
In the United States at least the subsidies for public transit pale in comparison to subsidies given to drivers in the form of road maintenance and parking subsidies. Subsidized public transit makes a ton of rational sense. The fewer cars there are driving around me the safer I am and the cleaner my air is. I want everyone to have an option to get around that doesn't require a huge upfront investment. It seems you and I are optimizing for different values but there are plenty of rational reasons to support it.
As far as convenience goes self-driving cars should be extremely convenient once we have them, no? I agree with you that public transport makes sense in very densely populated cities but most places in the world are actually wasting money on lots of public infrastructure that makes little rational sense, simply because it's a popular demand.