I'm an ex-Big Four Director (I now work in a very similar role - minus the sales targets - at a former client).
I enjoyed my time in consultancy as I'm used to working in a very, fast-paced aggressive environment anyway (investment banking). But the hours were brutal, the clients demanding (they love to play the consultancies off against each other) and usually the work was complex and challenging (no-one is going to pay for consultants for straightforward work - it's either complex or failing or both). Everyone judges you all the time. Constantly. My average hours were 12-16 per day. 12+ hours for my client (in their offices) then my additional consulting practice work (bids, recruitment, line management, sales, white papers, training etc). There can be a lot of travel (to some very humdrum places), often at very short notice. You may have to work away from home for 6+ months. New hires at Director level tend to be sent to work away somewhere 'grim' to cut their teeth. Whereas the existing Directors will have already done their 'grim' stints and proven themselves. This is actually one of the reasons why Directors tend not to move between consulting firms - no-one wants to prove themselves all over again as they know what a nightmare for your personal life that can be. As for flexibility, it depends on your clients. Some are nice people, some are arseholes (or nice people under huge pressure from their own management). Assume zero flexibility and just be happy if it's not too bad.
I came close to burning out several times. I finally decided to get out when I realised how warped my work-life balance had become. I called one of our salespeople at 11pm on a Mon night without even considering if it might be late (I was buried in work and it never crossed my mind). He answered of course and we had a normal conversation as if it was 11am rather than 11pm. Another member of the management team called me at 9am on a Sunday when I was on holiday (I was away and actually driving to a nice cafe for breakfast at the time). He didn't even say good morning but instead just launched into a tirade about a work issue. One of my directs asked me how on earth they were supposed to fit their mandatory training around their long client days and I replied "that's what weekends are for". All of this was completely normal in my firm. I never gave it a second thought. We were just consumed by work. And then it suddenly hit me that this wasn't normal (!!!) and I decided I needed to get out before I lost my personal life or sense of self completely.
Not everyone will have had this experience but this was mine. I miss it at times. But I wouldn't take on a role like this if I had children or a partner who wanted to actually see me. Partners have to be very understanding.
I earn less now but I'm 100% certain that my average hourly rate has increased.
Btw, look at their sick leave policy. Some of the consultancies are notoriously cheap with sick leave (if you're not billing, they're not happy!) and may only pay a max of 4 weeks full salary. I'm guessing that may be rather different to CS?
To summarise, you will get a salary bump but by God you will earn every extra penny. Nothing comes for free. Don't fool yourself that it does.