Or is it ok for me to only have contacted them without having a date set now?
You need to set the date.
The requirement is that you contact the EC by the 10th. So, I guess you could send out an email at 11:59 and meet the requirement.
However, do realize that many students wait until near or at the deadline. ECs are like most people at this time of year; busy. The later you wait (and you are up against the deadline right now) the harder it will be to find a mutual agreeable time for the interview.
The best interviews are where the interviewer is in a good mood and not feeling rushed. It also says something about your organizational skills of you are scheduling the interview st the last minute without a good reason.
I have contacted them a couple of days ago they just haven’t responded yet which made me wonder whether I need to have arranged the interview by the date
I had the same issue; as long as you reached out before the deadline you’re fine.
You should set a date because if you wait too long chances are good you won’t be able to schedule one.