You are in an enviable situation if recruiters are looking for you in this tough season when organisations are retrenching. My advice would be that you take part of your annual leave on the day of the interview to avoid lying to your employer since sick off should only be taken when you are indisposed.
You have no obligation to inform your employer what you will be doing when you are on authorized leave.
You could also consider bargaining with the prospective employers for an early morning interview or a session during lunch hour, this way, you would not be abusing your current employer’s time.
You could also have a preliminary telephone interview to gauge whether you would be interested to proceed to the next level of interview.
The recruiter would also decide if you have the skill set, attitude and competencies required for the available role. With the busy world we live in, sometimes it is not feasible to have face-to-face interviews. Time differences for candidates and recruiters who are in different countries can also be a hindrance, and in comes technology, which has enabled recruitment without the recruiter and interviewee being in the same room.
If the recruiters are keen to have an interview session with you, they will create time and consider your availability and convenience especially if they are the ones looking for you. Also, their inability to accommodate your request to be interviewed at times convenient to you could be an indicator of the kind of employers they would be.
The fact that they are very rigid also tells you what kind of an environment you will work in should you take the job. Imagine that you are an employee and are unable to attend to your duty early morning due to an emergency, would they allow you to?