Kirk's problem is that he promoted an ensign to chief engineer so they're all off doing something else... evacuating the ship maybe? And it really isn't clear why they have magical glass and inspired transporters but no radiation suits or robotic drones - technologies that we have in the 21st century already, along with greater equality of the sexes.
If there were radiation suits, he probably didn't have time to suit up. I doubt a radiation suit would've helped, even in TWOK Spock didn't bother with one did he? Everyone in engineering appeared to have one on, as far as I'm aware there has never been such a suit in Trek.
The Enterprise crew refused to evacuate, this was made clear when Spock gave the order to abandon ship. I'm not sure on your first point either, he had Scotty and the acting engineering officer at his disposal, would he have ordered them into the warp core to die?