All the chemical and biological (bacteriological) parameters should be evaluated, and the values should be within the WHO drinking water standards or any regional drinking water standards. Chemical parameters like pH, iron, fluoride, arsenic, etc., can also cause health problems other than bacteriological contamination.
I would say that total coliform is more important for drinking water use.
The US EPA has used total coliform for regulatory standards for drinking water. It suggests contamination by fecal matter, which would indicate a risk for pathogens. However, it is not as good of an indicator for pathogens as fecal coliform, as some coliform bacteria can grow in non-fecal material.
Total plate count indicates all bacteria (and other organisms), so it is not a good indicator of fecal contamination or potential for pathogens. It is, however, an indicator that there is some bacterial or organic matter contamination. I don't think health agencies typically use total plate count as a regulatory criterion.
I cannot speak in general, but to the best of my knowledge in Israel total bacteria counts is used (only) to estimate chlorinaction efficiency, not water quality. Water quality is done by fecal coliforms.
Talking about coliforms (fecal or total) did any of you looked into the origins of these assays? They seem rather general and are very hard to modernize, as there is no real definition of what species are included.