Can anyone help?

I would advise not using recruiters for the top-tier consultancies. IMHO approaching the firm directly shows initiative and demonstrates that you've researched the firm. Also, if you are for example a career changer, it allows you to sell yourself, whereas a recruiter would not forward your details because of your lack of strat consult experience.

If you're represented by a decent recruiter, which if they're recruiting for that client list they should be, they will be able to give you more info on the company to see if it's a good fit. A recruiter will also sell you to a consultancy and, again, a decent one will forward your details if appropriate, but won't if not. If you're an experienced hire it should be worthwhile going through a recruiter, but don't approach them. Make yourself visible and if you're relevant you'll get a call.

Make yourself visible?

Yes, if you're an experienced hire. Presenting at industry events, publishing articles etc. If you're a well known name in your field of expertise you're more likely to get a call re a position at a top company. Standard isn't it?