Op I think you need to ignore everyone’s opinions and ask the day boarding schools you are considering if they will put you in touch with current parents (some schools will do this and others not) and having spoken to them go with your gut about what will suit your particular dc and your particular job and home circumstances. If the school won’t arrange this, ask on a local Facebook group and speak to parents directly.
Families and dc are not all the same. And some people do (thank heavens for society) very important jobs. If we want female equality we need women to be able to do important jobs as well as men.
It’s always tough on the dc but the people saying “find a lovely nanny” are forgetting that finding and retaining the right nanny can be extremely difficult. It is brilliant when it works and you have continuity for many years , but you are relying on one person so if nanny is ill or absent or leaves suddenly for some reason you are stuck, and you are handing over a phenomenal amount of trust to one person without much outside policing (ie no other colleagues there to observe and verify what they are doing). Having a nanny can be problematic too.
And I say all this as someone who believes in making it easier for parents to work, as many EU countries do, and having destroyed my chances of having a proper career in favour of my dh’s high powered job as we felt there needed to be one person available for dc on their terms and not those of a society that’s attuned to commerce and consumerism.
Looking back I was quite depressed during the latter periods of being sahm and pt working mum though so the truth is that that isn’t always the perfect answer either! I am much happier now I can work more!
Finally I think a crucial element in all of this is how well your dc cope with having to have their “in public” face on all day as opposed to being able to chill at home and in private. Some dc have the strength of personality that they can find inner relaxation wherever they are. Some are more sensitive and need more of a bolt hole.