I would say it very much depends on what type of school you wish to work in. Some schools employ teachers without qualifications and others insist on the Diploma. I think the same could be said for promotion to Director of Studies level; schools have different criteria.
It will also depend on the type of tasks the Director of Studies is expected to fulfil. Tasks can be very varied: testing students with a view to placing them in the appropriate level, marketing the school and courses, training and supporting teachers, recruiting, timetabling….. You can see that some of these tasks can be done by experienced teachers; others such as timetabling and marketing don’t require any experience of teaching at all.
I suggest you get as much experience as possible in the above areas and any others that you feel are relevant. You could ask to be involved in such tasks in the school where you currently work. This would also send the message to the current Directors that you want to advance and that you are willing to take on responsibility and learn. You could also read books on EFL management; e.g. Ron White.
If you have a college degree and many years experience, you are part-way there. I think a qualification in TEFL helps; you say you are qualified but not what your qualification is. A qualification gives extra credibility in the eyes of an employer and fellow teachers.
I hope this will be of help.