COUNTER ARGUMENT TIME!
5. Elizabeth is not allowed to fight because she probably doesn't know how to use one safely.
4. She was in a contained environment. And if she tried to escape without Booker she would be caught and put right back in.
3. Maybe they don't overlap because they had only a minute or two on screen?
2. "Her influence is weaker than Bookers" There you go.
1. No, he shouldn't have. You want to make your own character? Play Fallout 3/NV or skyrim/morrowind/oblivion. And as Jim said, "Male characters make money". Does this make the original Bioshocks sexist? Or Gears of War? Or Halo 1-4? No. Also Booker wouldn't have been able to make the same connection or relationship if female without being homosexual (I have nothing against homosexuals, I think its ok to be homosexual because we are all created equal).