I guess Starfleet considered the viewing of classified intel more of a security risk than self-detonating a starship.

Well it is also a bit weird with these passwords and voice recognition to activate the self destruct of a star ship.
In ST-II we find out that to activate the presentation of that Genesis device, voice recognition and retina scan is required, which is a much higher level of security.
One would think that an equal level of security would be there to activate the self destruct.

Maybe there was an alternate destruct code that could have been used, much like an alarm panel's distress code, which would deactive the alarm but also notify the police. In this case, let's say that Kirk used 22b3c instead of 11a2b, it could have permanently disabled, or, made stopping the auto destruct sequence impossible.

The destruct sequence scene was keeping with continuity with what was shown years before in the classic episode, "Let That Be Your Last Battlefield."
Solarsailor is right that it's also voice recognition besides the passwords. Here is a clip on Youtube from someone who put both destruct sequences back to back and you can see in the classic episode it's voice recognition too:

Maybe that is just how it is to be worded? I really think you are over-analyzing this!

I am not a fan. I just happen to enjoy movies. Fans are embarrassing.

Then why did they have to identify to the computer who they were? Seems like the computer didn't recognize them otherwise

Voice recognition and such are added security. Not just anyone can self-destruct the ship with just the passwords.

I am not a fan. I just happen to enjoy movies. Fans are embarrassing.