My wife went back to her hometown in southern Taiwan a few years ago to help take care of her ailing father. One day while at the hospital she saw a tour bus pull up, and forty tourists from mainland China disembark. Their tour of the local sites included a checkup and some tests by the doctors in the hospital. Works out well for them and the hospital, which is not nearly as busy as the facilities in Taipei.
People clogging up emergency care facilities with common colds and minor ailments is common (as noted by others on this thread). There are several negative side effects including patients expecting to be prescribed something even if it's not warranted. If it's antibiotics, that's a problem because people can build resistance to antibiotics. I also believe there can be a kind of assembly-line mentality taking over among the staff at some busy facilities, with doctors spending only a few minutes with each patient. This can unfortunately lead to truly serious cases being overlooked.
For chronic conditions, the quality of care varies quite a bit. One complaint I've heard several times is sometimes there is no coordination between specialists -- it's up to the patient or family members to reach out and make appointments and communicate test results and the conclusions of other doctors. Lots of things can go wrong, and I fear some patients fall between the cracks.
Dental care can be very good and very inexpensive - my dentist here in the U.S. has complemented the skills of the dentists back in Taiwan who put in crowns years ago.