I come across alot of of PE pitchbook and memorandums in my job, and virtually all the contacts, people referenced, emails, etc. do not have any sort of CFA designation.
PE is more of Investment Banking. Basically sales + technical aptitude. In other words, identifying opportunities and making something out of it. This involves industry/company selection to capital structure optimization/exit plan (ie. how does the PE firm maximize its return).
The sales parts includes having strong soft skills and convincing everybody around you on the deal. Seeing that most shops are smaller and run really lean, you’d better have demonstrated in your past the ability to produce, or have somebody high up in the company really vouch or have faith in you (or you can be an Ex-President who can open up business opportunities and get the firm into contact with the right people)
If you are introverted, or hole yourself up studying for the CFA for several years, and not work on getting ‘that’ opportunitiy on the right track out of school, and not work on developing your soft skills and contacts, chances of getting into ‘PE’ from primarily the CFA is very low IMO.