The word and concept a lot of people seem to be struggling to find is 'Infrastructure'. To partially quote Wikipedia "... the fundamental facilities and systems serving a country, city, or area,including the services and facilities necessary for its economy to function". If you do not believe that Railways form part of the National Infrastructure I refer you to the assessments of the impact of the closure of the closure of the rail line at Dawlish Warren.
Of course elements of infrastructure can be eliminated and the impact worked around, but there will be impacts. For example I live on a boat, a large number of other people who live on one operate 'Off Grid', that is, without connection to the national 230V electricity infrastructure. They also live without landline telecommunications, another element of infrastructure. I myself live without mains sewage.
People can avoid using the railways but there will be consequences. To take a personal example - as I am not currently in a legal or financial position to move I would lose my job and have great difficulty in finding a new one as my skills are rather specialised and I have health issues. Thats a personal impact, but the national impact is that it would add, at least temporarily, to the Social Security budget and deny the MoD access to a Defence Analyst with 30 years experience and high quality Software Design skills. Repeat that many times and the problem becomes less than trivial - it can be worked around but it will cost time and money.
How you supply National Infrastructure is an issue for debate, but, generally, supply it you must. Remember that even that most automobile and Private Enterprise focused society, the United States of America, has a rail system, including 'local' commuter systems that rival our national ones. They are there for a reason (or, rather, a whole set of reasons).
As I have said, how you supply infrastructure is an issue for debate and analysis. Nationally and worldwide various approaches are used, there are a lot more issues involved than just 'Private' or 'Public' - there are many variations of 'Private', 'Public' and combinations of the two. Each approach has its advantages and disadvantages. For any given situation there will usually be at least two reasonably practical, worthwhile and optimal approaches - which one you choose is then down to which advantages and disadvantages you prioritise, or even which you perceive as 'advantage' and 'disadvantage'.