Law in AZ: to require new voters to provide proof of citizenship when they register. Law was passed a few years ago but was prevented in court from being enforced. SCOTUS said it may go into effect. 5-4 decision. (Should have been 9-0 in my I opinion)
Birth certificates are often handled at the county level, not the state level. Not all counties offer birth certificates by mail and not all offer it under all circumstances. For instance, people with name changes and those who have been adopted don't qualify for remote processing in the jurisdiction I referenced. Therein was the issue. He didn't know it at the time, but his mom changed his name a few days after he was born. Hence, he had to drive to get his birth certificate. I am familiar with how nice it is to be able to get copies online. I've gotten my own and my mom's that way.
The mail issue largely relates to duplicate voting... and the predominant reasons that happened is because 1. A lawful voter forgot they voted by mail and then votes again in person, or 2. A lawful voter verifies their ballot hasn't been counted and goes in to vote in person, only for that mail-in ballot to be counted later, after they've voted in person. These aren't issues with people fraudulently registering to vote. Again, if they were, they would have been included in my counts.
As far as the homeless argument... I didn't say all homeless people vote democrat. I also tried to find a single source that backed up your claim about illegal immigrants taking up the space in shelters and forcing Americans out. I couldn't find anything on that at all. The closest I got was a a few reports about how temporary shelters set up specifically for refugees and asylum seekers were at capacity. For clarification, refugees and asylum seekers are here legally. They're not undocumented. And, these reports talk about how they're using services specifically designed for them. Of course, how that gets reported varies depending on the outlet reporting. The Heritage article, for example, talks about a shelter in New York that was set up for asylum seekers that had to turn people away. But, rather than making it clear who that shelter was for and who those people were, it moved into a discussion about illegal immigration, as if undocumented immigrants were to blame for these facilities being at capacity and those shelters were for everyone to use. They aren't the same thing and those shelters were not for Americans. But, you wouldn't know that by reading the Heritage article.
Without any evidence of "homeless being forced out of shelters to be replaced by illegals," it sounds like this is just another scare tactic being used - something that isn't really a problem being exploited like it is a problem so that legislation gets passed to address something else... much like how AZ's law to prevent "illegal voting" isn't really doing that, but is skewing elections in the favor of republicans instead... though it sure sounds nice to think it's doing something about illegal voting. Kind of like your question is posed - who could possibly be against a law that prevents illegal voting? Underneath it though, when you know the data, it's slimy.
I don't want to go on a rant, but really, this is what politics has become. It's cloak and dagger. Smoke and mirrors. "Pay no attention to me while I'm making sure my pockets stay full and the disadvantaged stay that way. Look over here at this really terrible non-problem that I'm fixing."
Are we reading the same articles? The one you just linked explained that "it is department policy to transfer those in shelters based on a variety of different needs, including capacity, building issues and need." It also only talked about one family being moved - it was a migrant family that moved to a new shelter because they felt unsafe. This article is also clear that it's referencing asylum seekers, not illegal immigrants.
I'm not sure what site you got your data from in terms of voter issues. I got mine from Heritage Foundation - there were literally only 32 total cases listed in Arizona (a 3-page list), of which 1.6% related to ineligible people registering to vote.
If we're going to require documentation for someone to use their right to vote, we should make it free and easy for them to get that documentation. Voting should not be a pay-to-play game. That's neither fair nor equal.
As for birth certificates, I'm not sure what your point is. I acknowledged it's typically easy to get one by mail, while also citing a personal experience in which it was not. This wasn't hearsay. This was a situation in which I personally believed the person was full of crap until I tried to help them and realized they really did have to drive to their location of birth to get a birth certificate. Unless you're trying to tell me that my story never happened, than any other explanation you might have about how you can get them by mail is a moot point.