How often could a mission to an unknown, inhabited planet be considered "not diplomatic"?
Even if it was known in advance to involve fighting back angry hordes, Kirk beaming down with two redshirts armed with phasers would be equivalent to MacArthur wading up the shores of a Pacific island with the entire 3rd Marine Division
going hut-hut-hut around him. Hand phasers aren't honor guard sidearms, they are strategic weapons! Generally speaking, landing party duty should be rather risk-free. The problem isn't with the demonstrated practices of beam-down, it's with the implausible calamities the parties are subject to.
As for sending down the CMO, she doesn't run a hospital full of doctors. She is one of perhaps four MDs aboard (see "Remember Me"), and that's pretty much it. One of the four has to go down whenever mere paramedics don't suffice - and the CMO isn't a poorer choice than the other three.
Sending the CEO out in "Samaritan Snare" was just plain silly. But perhaps it was diplomatic?