What would a career coach do differently?
Not so much what they can do but what they can advise/what sort of feedback they can give - if it had worked then why are you still in this position? Multiple people have interviewed you and there seems to have been a problem.
Anyone can conduct a mock interview, maybe it will be useful in and of itself for the sake of simply practicing and maybe they can give you some feedback - I remember people would do that at school and uni (careers service would offer it etc..).
A recruiter or former HR professional might well have a bit more insight.
The bottom line is you've got a problem that you're apparently unable to fix by yourself ergo you need feedback, short of recording yourself in an interview and posting it on here none of us are able to give you much feedback beyond guessing based on stuff you've posted ergo you need a third party to do it.
I've not tried toastmasters but just googled it. Looks like it's American.
So what? Is there something wrong with organisations headquartered in America?
There is a guy currently doing it in this thread, I know people who've found it very useful:
https://forums.overclockers.co.uk/threads/toastmasters.18949603/
Failing multiple interviews you're otherwise suitable for on paper does suggest soft skills could be an issue. It might not be that, there might be some domain-specific things you're tripping up over or some general competency questions, an ex recruiter won't likely be able to help with the domain-specific stuff but can pick up on general soft skills issues and competency questions. You could try to find former uni mates/colleagues to help with domain-specific questions.