Although I agree on the silliness of the citizenship-at-birth rule, a maximalist approach to democracy can be very dangerous. The most unstable democracies of the past century were, often, the most democratic ones - Weimar, various French republics, etc.
The democratic paradox is real, and finding ways to minimize its worst outcomes can be legitimate.
I agree that this is a valid concern. However, we should then carefully review which restrictions actually help with stability and which restrictions are a nuisance.
It's instructive to compare the constitution of the Federal Republic of Germany with the Weimar Republic. For example, while modern Germany still uses proportional representation, you need 5% of the votes to get any seats at all. (I'm simplifying a bit.) And you can no longer have a pure 'vote of no confidence' in parliament to bring down the government, you need to simultaneously put a new one in power, see https://en.wikipedia.org/wiki/Constructive_vote_of_no_confid...
As for the US president: I think the requirement for a minimum age and for citizenship are fair enough, because these are fairly easy to verify once and for all. But I think that the requirement for citizenship at birth is, if anything, bad for stability: remember the birthers?
Now imagine that in Obama's 6th year in office, some random birther had actually found some reasonably compelling evidence (but not compelling enough to make even Obama supporters agree). Can you imagine the chaos?
The democratic paradox is real, and finding ways to minimize its worst outcomes can be legitimate.