Yes, the Timetabler is usually a Deputy Head or a Head of Department [often Maths or Physics]. Occasionally it's done by a team of people, which has advantages & disadvantages. Timetabling is mainly about problem-solving but it's also about diplomacy & negotiation, so the Timetabler needs to have some status in the school.
There's a myth that big schools are always harder to timetable than small schools. Big schools have much more data to handle, but small schools may have a disproportionate number of part-timers, and staff teaching 2nd/3rd/4th subjects.
And just because the timetable was easy (or hard) last year, doesn't mean it will be the same this year [it depends on the conflicts caused by the particular arrangement of staffing each year]. That's all part of the drama (and fascination) of timetabling.
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