Failed to get this simple sentence

suc·cumb
səˈkəm/Submit
verb
fail to resist (pressure, temptation, or some other negative force).
"he has become the latest to succumb to the strain"

synonyms: yield, give in/way, submit, surrender, capitulate, cave in

"she succumbed to temptation"
die from the effect of a disease or injury.
synonyms: die from/of

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in the given context I'd consider surrendering or yielding as good alternatives. If I was trying to reach a wider reading audience, I might use falling instead.

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stake

http://dictionary.cambridge.org/dictionary/english/stake

have a stake in sth
› If you have a stake in something, it is ​important to you because you have a ​personal ​interest or ​involvement in it:
Employers have a stake in the ​training of ​their ​staff.

stake noun [C] (RISK)
› the ​amount of ​money that you ​risk on the ​result of something such as a ​game or ​competition:
She ​spent two ​weeks in Las Vegas ​playing high-stakes ​blackjack at the ​casinos.

raise/up the stakes
› to ​increase the ​prize or ​reward in a ​competition or any ​activity in which you are ​competing
› to make a ​situation more ​urgent or more ​difficult to ​ignore:
The ​stowaways are ​trying to ​raise the stakes by ​refusing to ​eat until they are given ​money and ​aid.

stake verb [T] (RISK)
› to ​risk an ​amount of ​money:

In the given context, I'd consider using risk.

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http://www.merriam-webster.com/dictionary/survive

: to remain alive : to continue to live

: to continue to exist

I'd use something like exist in the context given.

Our world is surrendering to the activities of men and women who would risk the future of our species on beliefs that should not exist in an elementary school education.

or perhaps

Our world is surrendering to the activities of men and women who would risk the future of our species on beliefs that should not be part of an elementary school education.

not sure why you'd want to change it though - it's clearly written

The idea expressed in the phrase "beliefs that should not survive an elementary school education" may well refer to beliefs so irrational any elementary school student should see through them.

Perhaps the author equates faith with credulity.

Thank you both