I wouldn't call it a "careless attitude toward security". She's thinking in a different way than we do, because she's a diplomat, not an engineer.
There is no technical reason that a privately administrated email server would be inherently less secure than a government-administrated server (there are good arguments that it's likely to be more secure). However, a private email server is likely to be far more user-friendly and free of "security theater" constraints. Speaking from experience, the usual approach of government and other large organizations to "security" is to throw user experience out the window, forcing ugly/retro "proven" tech on users, requiring complicated and difficult administrative steps to use the system, slow approval and ticketing processes, etc.
The primary job of the Secretary of State is to communicate. Any time wasted on arbitrary tech hoop-jumping, any restrictions on how that communication happens, is keeping the SoS from doing their job. Can you imagine if we were in the middle of a political crisis and suddenly the Secretary of State is on hold with tech support while dealing with a forced password reset or something equally stupid? American lives at risk, and Lotus Notes is the only way to communicate? Etc. See the issue here?
To really resolve the problem, they would need a relentlessly service-oriented approach for whomever is responsible for email at the State Department. It would have to be as friction-free an experience for the user as possible, within the boundaries of security.
Until then, every Secretary of State is going to put their ability to communicate quickly and easily with the most important and powerful people in the world ahead of the kinds of technical wank that the average HN user thinks is important.