Why consulting rather than private equity?

Generally, PE and HF's are considered the apex of finance. Very few would take a big name consulting firm over a big name PE firm if their interest is to become a professional investor.

There is a supply and demand issue - the supply of many PE roles is smaller than consulting roles, so many folks target consulting roles over PE.

That being said, here are a few reasons one may take consulting over PE.

1. You have little to no interest in being a professional investor. The exits from consulting are broader and include marketing, ops, tech and other functions/roles with it's own corporate ladder. If you want to be a BU head at a port co, PE is a better fit. But most PE port co's are mid market; if you want to be at a market leader (Disney/JP Morgan/Google/Kraft/etc) than consulting is a better resume stamp.
2. A good consultant might not be a good financial modeler. I know plenty of folks that stick to consulting because they don't have the modeling chops nor the desire to be as excel driven as PE requires you to be.
3. Not all PE firms are created equal. Most PE activity is middle market. There's huge variability of firms within that space. Believe it or not, consulting can be better job security than most mid market PE. If you're counseled out at your consulting firm, you have an enormous amount of resources to keep you on your feet. If your whole PE fund goes under, it's more of an uphill battle.
4. Big name consulting has better brand recognition than the vast majority of PE firms.
5. PE requires a very long term time commitment if you're gunning for carry. Most people join consulting to try a lot of different things for 2-3 years before they exit.
6. Usually it's consulting THEN PE; so it's not necessarily a mutually exclusive comparison.
7. The path to partner is clearer in consulting and often fairer; in PE, you have to wait for a whole fund cycle to typically get promoted. In consulting, you get benchmarks every 6-12 months indicating how close you are to the next level.
8. The feedback culture at most consulting firms is second to none. Most PE shops just expect you to perform and cycle you out if it's not working out.
9. The majority of PE roles for junior staff are pre-MBA, with the knowledge that you will get pushed out after 2-4 years. Once out, it's very difficult to find another job in PE, including a lateral. Stepping up a role is even more difficult. If you're pushed out in consulting, lateral opportunities are abundant, and industry roles line up to recruit you at similar pay.
10. PE culture can be as toxic as banking at many funds; since most staff are ex-bankers. This is less prevalent as consulting. It is also an old boys club, which can make things difficult for woman and minorities, and it's not uncommon for a 25 year old associate to just reek douchebaggery to new hires. Note that I am not generalizing the industry, but this culture definitely exists in greater mass than in consulting.