I agree that one board would make a lot more sense – it would also save some bother when (if) students move schools mid-GCSE. As others say, the main reasons are historical. I took Cambridge board O-levels (old) but for a couple of mine we did AEB as it was reputed to be easier. I am not sure if that is what moves schools to pick one board or another.
I have always taught AQA for my subject (MFL) but there is also an Edexcel option; tbh it is pretty similar so not much to choose. It might be that a school has the AQA textbooks for example, so why change? IGCSE MFL is very different, not least bc it is intended for a range of native tongues (rather rather than only English speakers) so the entire paper is in target language and there is no translation into or from English (for obvious reasons). For that latter reason I would always advise taking GCSE rather than IGCSE in MFL, as I think translation is a key skill.
Re English and @tennissquare whose son’s private school does another board (which one?) which offers more options than Macbeth and AIC – of course there are many other options on the AQA syllabus too. There are about half a dozen Shakespeare plays to choose from and a dozen or so 20th cent texts, including plays, novels and short stories. Obviously schools tend to choose to teach the same texts for the reasons given by @theresnolimits. Makes sense really. But my DD did R&J not Macbeth and Animal Farm not AIC, even tho she did AQA. Also exams are no more spread out at the private school which chooses other boards than they are at any school. Exams have been more spread out in the last two years as a Covid concession (in 2019 DS2 had up to 10 in a week).
AFAIK only Cambridge IGCSE has exams at a different time – otherwise pretty much everyone who did (for example) French GCSE or IGCSE, AQA, Edexcel, Eduquas, Pearson IGCSE, sat an exam or exams on Tuesday morning. The various boards have to schedule at the same time or there would be endless clashes.