Because your right to vote is fundamental, and shouldn't be taken away just because you lost the paperwork.

Plus, non-citizens understand that trying to vote will ruin any chance they may have in the future of getting citizenship. People here illegally also don't want to call attention to their presence here and won't risk trying to vote based on that. Those people are not voting in any meaningful capacity.

The voter suppression thing is real, though . Click that link I left above; in certain states, there is a regular purging of the voter rolls, for frivolous reasons. Some voters don't find out they have been purged until they show up at the polls (after waiting in a long line to boot). Not giving them some way to cast a provisional ballot is the same as disenfranchising them.