OK let me try and tackle this one, as the author of that particular chapter - and also give you some constructive suggestions.
Firstly put yourself in the shoes of a FTSE100 client paying a strategy firm £1,000-£1,500 per day for a junior consulting team member to work on their project. They engage the leading brands in part because there's a quality threshold that they bring to the project. What these firms want to portray is therefore a uniformity of excellence. When the client looks through the CVs of those who are going to be on their consulting project team, there shouldn't be any CV that gives the client cause for concern.
Whether or not that concern is valid, the issue is that top consulting firms have enough candidates who offer exactly what they are looking for that they don't need to "take a chance" on someone who - on paper - looks like they don't fit the mould.
There are two possible ways around this. The first is to target only very small niche consulting firms - those that most people would not have heard of. These firms tend to sell clients on their expertise and / or the uniqueness of their teams. Hence for these firms there is less likely to be a hiring mould that they simply cannot diverge from. For these small firms, where everyone has to get involved in every aspect of the business, your business development experience could be a strong selling point.
At some future point in your career this could open up the chance to move to one of the mainstream strategy consulting firms - as it's usually what you've done in the last years that counts the most on your CV, so your unorthodox years post-graduation will count against you less and less over time. Or you may decide you like working for smaller more entrepreneurial consulting firms and have no reason to want to move to one of the big names.
The other potential route would be for you to gain an MBA from a leading business school (only bother with the really top schools though). Strategy firms will hire from these schools candidates who do not have prior consulting experience, so there's a possibility this route could pay off for you. Have to say I've not got the exposure to the business schools to know whether they'd take someone who's been working in a small family business. Certainly LSE would put you in with a shot if your grades were OK.
Or I guess you could combine both - small niche firm followed by top MBA followed by leading strategy brand.
Hope this helps and good luck