The reason for (some) separate comps in chess is the same reason why we have scholarships just for women in STEM subjects. It's to encourage female participation in areas that women have traditionally been underrepresented in. Interesting that the BBC find 3 people to complain about the ban but don't report those chess players who are in favour. Nobody has suggested it's because women's brains aren't as good at chess.
As to Park Run, men have an unfair advantage in these events, regardless of how they identify. Scientific research shows this conclusively. Whilst Park Run is presented as a 'fun run', they record personal times and course records. The organisation itself ranks runners and publicises and celebrates these times and records. If an event is going to record times by category (Sex, Age or whatever) then those categories should be fair.
Would it be fair for a 25 year old woman to have their results recorded in the Women's senior category in Park Run, beating everyone in that group easily? Of course not. PR even have separate graded results which account for age. You can compare your result with a figure that balances out age advantage. But they let men simply self identify straight into the Women's category ...
There are PR Women's records that are now held by men and are unlikely to ever be broken by a woman.