Except universal suffrage is a relatively new concept. Democracy existed before it. I'm not saying it's the right path, I think it's silly to consider such a thing, but I also think it's silly to think it'll necessarily lead to widespread suffering, at least significantly more than we already have.
Citizenship birthright is not about the vote, that's the least part of it. Read up about people who never got a birth certificate for whatever reason and the difficulties they experience. Generally it's illegal for them to drive or have a job, amonst other things. And that's actually people who nobody actually doubts they were born in that country, they just don't have any paperwork. It's also illegal for you to leave the country (because you can't get a passport without proof of citizenship), so you can't even escape to somewhere else. That's what being a non-citizen is like. Also, can you even own a house if you can't prove you are a citizen? What are the laws on home ownership for illegal immigrants? I'm guessing they cannot own a house, or have a regular bank account. All those things require verified government documents.
Things like the Bill of Rights and constitutional / legal protections tend to only cover citizens. So if you start growing the proportion of non-citizens to be a significant proportion then that's a new class of Untermencsh who have no legal protections, yet no other country to go to, so they're at the mercy of whoever wants to exploit them. And someone
always exploits those who are weak. You don't live without your basic "rights" afforded to you by citizenship, so you have no idea how fucked things become for people who don't have them.
There are somewhere between maybe 10-15 million undocumented immigrants right now, and if you took away the automatic citizenship birthright, you'd end up down the track with maybe 20% of the country being stateless individuals not covered by constitutional protections. And I'd hazard a guess those stateless people would be poor, and therefore also have a higher average birthrate than the citizens, so you'd have a situation where the proportion of non-citizens would tend to increase over time. Hence, what I mean by goodbye to
modern democracy. It would become akin to Athenian democracy where only a small number of people had basic rights and the rest were a slave caste.