The problem isn't getting any old random ID. It is getting one which prove you are a citizen of the United States.
If you have never gone by any other name, and your birth was recorded as a matter of routine - or you were a naturalized citizen who has never gone by any other name, it is probably relatively easy. You get a copy of your birth certificate or a copy of your naturalization certificate and present it.
But for around half of the citizens, it's not that simple. Take the scenario where you marry and change your name, get divorced but keep your name, remarry and change your name, divorce, revert to your first married name (since all your kids have that last name) and then remarry and change your name again. You have to prove that entire chain of name changes in order to prove your citizenship.
First you have to get a copy of your birth certificate. It may need to be a certified copy. You may be able to get it remotely - you may need to prove your right to get a copy depending on the state. $$ and time, depending on how long the process takes (going to get it, hiring an attorney to get it on your behalf plus processing time).
Then you have to get a copy of your marriage license - probably certified. You may be able to get it remotely - and, again, you may need to prove your right to get a copy of it. $$ and time, depending on how long the process takes (going to get it, hiring an attorney to get it on your behalf plus processing time).
Then you have to get a copy of your second marriage license - probably certified. You may be able to get it remotely - and, again, you may need to prove your right to get a copy of it. $$ and time, depending on how long the process takes (going to get it, hiring an attorney to get it on your behalf plus processing time).
Then you need to get a copy of your divorce judgment granting you a return to your first married name. You may be able to get it remotely - and, again, you may need to prove your right to get a copy of it. $$ and time, depending on how long the process takes (going to get it, hiring an attorney to get it on your behalf plus processing time).
Finally, you need a copy of your most recent marriage license - probably certified. You may be able to get it remotely - and, again, you may need to prove your right to get a copy of it. $$ and time, depending on how long the process takes (going to get it, hiring an attorney to get it on your behalf plus processing time).
You may not be able to do these simultaneously, since you may need some of the earlier documents to prove your right to the later ones - and may need to obtain them sequentially.
This is a relatively common sequence, and not an unusual number of marriages and divorces these days.
Funding the police department to issue IDs won't diminish the number of documents (time, and money) needed to get an ID that will satisfy the SAVE act (or any other act requiring a proof of citizenship).
The burden falls mainly on women (the ones who generally change their names when they marry). It also falls on older, poor, and/or black individuals who are more likely not to have the documents on hand - or never to have had them because they were born outside of the hospital and their births never recorded. (My father (94) was born at home - fortunately his birth was recorded. Many people that age who were born at home don't have a birth certificate - particularly blacks, whose midwives were discouraged from recording them. Those that were recorded were sometimes destroyed (or the secondary proof church documents destroyed) in violence that impacted predominantly black churches and institutions in the civil rights era.
So no, it's not a solution - EVEN if the premise were true - that non-citizens were voting in any significant number.
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