EU countries try really hard to register all births with the proper documents that then lead you to get an ID or passport later on in life. Everybody is used to needing an ID, being able to show it upon request, and using it to vote as well. Most people are just fine with all of that.

The US doesn't follow this logic. For example, a person born earlier the last century in the deep south may not have received a birth certificate. The birth could have just been written into the household bible. So when you require a voter ID and to get that you need a birth certificate, you have eliminated a possible voter. And what if I added this bible practice was done predominantly in African-American communities. There will be other loopholes where people will be required to produce paperwork they can only acquire after a Kafkaesque tour through various administrations. Funny enough a lot of them affecting minority communities first and foremost.

If you want to do a proper voter ID, you need to first convince the people this isn't government overreach (good luck). And then make sure everybody has an equal chance to get this ID, e.g. by issuing missing birth certificates. The laws that generally want to introduce this do not care about equal chances. They tend to target voters who vote for the other side.