I think it's fine to ask for feedback, although if they aren't willing/able to share this with you then don't keep asking.
As a private organization, they are definitely not under any obligation to interview anybody (as long as they are following the law).
There could be lots of reasons they didn't invite you to interview. The most likely probably are:
-More candidates met the criteria than they could feasibly interview, so they picked the 5/10 who were most experienced/had the most relevant experience/had the most recent experience etc. I know you think this isn't likely, but they could have lots of applicants, especially if people are jumping ship from NHS to private or the jobs offer better terms/conditions/pay than the NHS.
-They received so many applications they didn't have time to process them all properly and just invited the first 5/10 people who seemed suitable to interview.
-They specifically want to recruit less experienced staff they can train to do things "their way".
-Your profile (more experienced/returning from a career break) may make them think that you aren't going to stick around/be happy in the role long term and they want someone who will be. I know you say this isn't the case, but the employer can never know if that's 100% true and they may have had issues in the past.
When you're used to getting employed it can be easy to take these things personally, but honestly, it's not a personal slight. It's just a business decision. Interviewing lots of people is hugely time consuming and costly, so companies do have to put limits on this.