The answer as always depends on;
- the quality of the state and private options available to you
- the level of financial sacrifice required to go for the private option.
So, if there are good state schools (and there are lots) and private schools that aren't great (and there are lots...and think carefully about how you are going to measure if they are good or not) then it may well not be worth it. Additionally, bearing in mind that fees rise much faster than inflation and jump steeply at certain key points, for many people, paying fees will mean a significant impact on their lifestyle - perhaps no holidays, perhaps having to live in a house that is really too small or living in fear of needing to buy a new car or an applicant's breaking down - in this case, I would say that the 'cost' is too high and not 'worth it'.
However, if the state options are truly awful and its not possible to get a better state option (and there are often ways - think of all the sharp elbowed middle class who manage it somehow) and the independent school really is fantastic (not just shiny but delivers a fab education in general terms or for the specific needs of your child) then it might be worth considering, especially if you are very well off and the sacrifices involved in paying fees are small.
As ever, there is never a straight forward answer - it all depends on the schools available in both sectors and the sacrifices required. There are dozens of these threads and the usual view towards the end of them seems to be OPs who recognise that paying the fees will be a struggle, if not at the start, very quickly.....when they understand the jumps in fees and the big increase at secondary, many decide to at least start the State option and to start saving hard and then to re-assess if the state option does turn out to be. Disappointment or to re assess after primary. This is especially the case for people who are only just considering it close to a child starting school, so who have not really always planned for private school and so put plans in place for the funding.
I would also say again, especially at primary level to consider carefully what 'better' and 'worth it' involves - often there is no data to from private schools and it's hard to compare outcomes academically. Small classes and shiny facilities don't necessarily make things better or worth it. And remember that outside London, most Preps are not academically selective but have a range of ability.