I think it is dangerous to generalise, however what you said (spend 4-6 years in a graduate programme in a blue chip), is exactly what I did. I believe I reaped the following benefits from this career track:
(1) I had a "name" on my CV and some business track record, both of which are attractive to consultancies;
(2) I had (have) more credibility working with clients because I had actually experienced real-life business processes (rather than just mapping them). If you have a chance one day, ask a senior manager who has used consultants before, what he/she thinks of newly minted graduate "consultants". I think the answer will be less than flattering.
(3) I could(can) deliver better results for my clients because I understand how things get implemented in large organisations. And that, you will agree, is that ultimate goal of consulting, to have your recommendations actually implemented (whether you do it or someone else). As an aside, I once worked with a *brilliant* ex-McKinsey guy on a strategy project who was surprised when I told him the CEO is not the only one who had to agree to the strategy we were proposing. He thought if the CEO said jump, everyone would ask how high. That might be the textbook theory but it is not how 99% of organisations work.
(4) My career progression once I joined a consultancy did not suffer compared to someone who joined as a graduate.
(5) I was able to move back to a corporate role after 4 years in a consultancy (albeit to an in-house consulting role). Some of my colleagues who had spent their previous careers in consultancies found the transition quite challenging.
My only caveat is that 4-6 years after finishing your degree is probably optimal to join a consultancy. I wouldn't wait beyond age 28-30 (as I've commented on elsewhere in this forum).