As a modern compromise, I think the US should allow people who moved to the US before a certain again (maybe 10 or 12) or have lived in the US for 20+ years. If you want to go a little further, you could require them to renounce any other foreign citizenships upon successful election.
The Australia law caused a bunch of trouble in the last 10 years because a bunch of MPs accidentally had US citizenship by birth to Oz parents living in US or one parent was a US citizen, but they never lived in US. (US citizenship is a bit viral in that sense!) I don't remember all of the details exactly, but it did make me think more deeply about a nationality policy for MPs. I think it is a reasonable requirement.