In my opinion students do not like to count because they are simply not used to it or do not know how to do it. I agree with some of the other posters that if you make counting engaging and musical then they will be more likely to see the value. Often I will say letter names in rhythm, clap, sing in rhythm, and have them echo or do it with me. Often for students who struggle I will just have them identify whether a pattern is long or short. What I tell students is once you get advanced, you play rhythm based on a feel or what some call an internal clock, anticipation where the beats are, and honestly just by remembering rhythm patterns from playing them before so many times. There are really not that many basic rhythm patterns. For me when I read music, most patterns I already know and the only difficulty I may have is knowing where the beats will land and where.
On a side note I think all teachers should at least introduce the metronome at some point and show students how to learn with it sometimes as well as without. Students are not born with a good sense of timing and although they may be taught today with good time without the metronome they need to familiar with how to play with one in order to be effective in slowing down difficult passages, setting goals for themselves, subdividing, and playing with other instrumentals. To avoid teaching how to use it will only short change them in the long run. I am not saying use it in every lesson but avoiding completely will not allow students to be as effective learning a new piece as they could be.