Personally, I don't think LinkedIn is particularly important for Consulting applications.
To each firm you will apply with your CV and Cover Letter. This takes precedence over anything else for your experiences. Obviously, your LinkedIn should align with your CV in terms of experiences (or this would raise some red flags), but recruiters/CV screeners are unlikely to check it at all (just think about how many applicants they get).
LinkedIn because much more important later on, to network and connect with headhunters, who can help finding jobs in other industries after Consulting.
My recommendation would be to keep it short, professional, without being over the top. Your CV and experiences will speak for themselves.