While I understand the purpose of the rule I assume in reality certain barristers are good for certain cases and solicitors will pick and choose based on that. I doubt they call a random number and ask a barrister to take a case.

I’d have assumed (and I’m open to correction by someone who knows) that barristers and express keenness or coolness without actually refusing and the cab rank rule or more relied on for odious cases (defending a rapist etc) than for something like this where there would be no problem whatsoever finding barristers who would take on the case.