Based on the comments in this thread it's pretty clear why: People move all the time, and the state does not necessarily have a current address for everyone.
If someone was to try and vote twice from 2 addresses they simply shouldn't be able to because of the unique identifier.
I mean, if they were registered to vote in 2 separate places, they could just drive to 2 different polling stations and vote twice anyway, right?
That's part of the whole signature verification thing that isn't working well.
Officially the post office is supposed to do return to sender if the name doesn't match, but I suspect in apartments with frequent turnover they can't realistically do that.
Having the voter request the ballot gets rid of all of that trouble.
Any kind of double voting shenanigans should be pretty straightforward to detect. So when it happens, you [at least] invalidate the second attempt, and forward the details over to the police for investigation and prosecution.
The thing about voting is that most of the legwork in verifying voters is done in advance of any election. By the time you get to ballot-casting, the hard part is done.
You need to find a systemic vulnerability, otherwise the risk is much too high and the odds of success much too low. Vote-by-mail isn't actually a worry unless you have an ideological position based on the current election, you should really worry about the electronic voting machines used in many places that have in-person voting. The potential for systemic fraud there is much, much higher.
The argument that Republicans want to "make it harder for poor people to vote" is a red herring regarding the voter ID debate.
Similarly, Trump's position on mail-in-ballots, as far as I can tell, is that that the expected ease of large scale voter fraud by relatively few bad actors will make it impossible to win. I have no clue if that's true or not, but to frame the stated claim without also acknowledging the underlying potential issue seems misleading.
Genuinely interested, as the comment seems to be accurate based on a handful of Google searches I just did.