Agreed. I may come across as xenophobic, or uber-patriotic, or whatever, but I think the laws regarding eligibility for presidency should be changed, not relaxed.
For example: Required to be a citizen and resident of the USA for the majority of his/her life (>50%). Required to renounce all other citizenship. Required to have spent the last 10 years of his/her life in the US.
Something like that seems fair(er) to me.
EDIT: Clarification.