It's all good, first of all i guess english is not your main language and you have a little trouble in expressing yourself (im just guessing),
is like you are talking in reverse or something, i will try to clear/understand what you said:
"Disable death and birth, with this checked"
- What do you even mean by this?
"The character, clan mates and NPC lord will NOT dead of old age or in live-combat(where player participates) battles.They also won't make any new babies."
- IF you meant having the box birth and death checked you will have those results that is false, that is exactly what having the box unchecked like you do does and not the opposite.
"With it unchecked lords and player can die of old age and can make babies who eventually become functional lords. NPC lords and clan mates have a 10% of dying when they take go down in live battle."
- Like i said before this is false, it's exactly the opposite of what you are saying.
"and cannot find anything that suggests that executions should only work if birth and death is on."
- I sincerely haven't searched or read about it so i can't confirm or deny
"and since this has worked fine up to 1.7.1, I thought that this is a bug."
- Same answer for this.
Im not a game dev or anything so i can only "guess" about why this doesn't work with the box uncheck (you can't execute), and that would be because they might not want players to just kill nobles left and right and then having no means to end the main quest and/or just end the game prematurely because of this mechanic, again im just guessing here.
Execution works with the birth and death checked because there is always babies coming up so the chance of the game ending because of executions is lower compared to if this was to happen with the box unchecked.
I hope you understand this, because the way you said those things above, you were explaining everything in reverse
Box checked:
Nobles/Companions die - Killed in combat / old age
Babies are born
Execution works
Box unchecked:
Nobles/Companions don't die
Babies are not born/made
Execution "doesn't work" does not apply the relation penalty and the noble doesn't die.
To finish this, you have been playing with the box unchecked and you expected to have the results from having the box checked so.... that's
not how it works. Now the bug IF any would be if executions were actually to work still with the box unchecked.