It's based on an ancient and now disproved belief. Talk about 'meridians' and 'channels' in the body are completely fictitious and have been throughly debunked by modern science.
It works (if at all) purely by placebo. While this can be great for people who are feeling a bit 'off', unsettled or have a general feeling of unease, it does not work for real medical conditions.
It can give people a false sense of hope when they are genuinely ill and looking for a cure.
Unscrupulous practitioners can lie and abuse their position of trust. With little or no regulation, reflexologists can do what they want.
While it is a nice foot massage, it is just that. Any other claims about it is a lie.
It's not used by doctors or in hospitals because it does not work.