You don't need an ID to vote in Australia either. It works fine. What's hard to imagine? You get checked off on a list which includes all enrolled voters for that electorate.
It’s not 100% foolproof, but it turns out voter fraud by impersonation is very rare, so it’s good enough.
When you think in terms of “make sure every vote we count was legitimate”, then “not completely foolproof” becomes a solid argument for voter ID.
Instead, if you take a wider view and think in terms of “getting the best quality estimate of the will of the voting population”, the argument against requiring ID (in the US at least) is that it would distort the results of the election far more than a tiny amount of undetected impersonation fraud does.
This will vary by country. In the US, there are barriers to getting ID for some groups (you need to go in person during business hours, pay and wait an unknown amount of time, and this needs to happen weeks ahead of election days; this is a barrier to someone without transport juggling multiple jobs and childcare, for instance.)
Other countries see the trade off differently, or use different fraud prevention approaches. For example, I know India uses indelible ink stains on fingers to prevent multiple voting, and in the UK, there is no ID requirement (yet) but the ballots are serialised and the secrecy of the ballot can be broken to investigate fraud allegations. Neither of these approaches would be culturally acceptable in the US.
Australians are also fairly trustworthy
https://www.jstor.org/stable/27550168
Personally I think voter ID makes sense even if only to quash allegation of voter fraud. Voter ID enjoys overwhelming (>80%) bipartisan support from regular Americans. The main obstacle to new voter ID laws is the Democratic Party establishment. They calculate a marginal decrease in electoral margins if new voter ID laws were to be enacted. Of course then they wind up faced with fiascos like January 6, but politicians are nothing if not short-term planners.
https://www.cnn.com/2021/06/25/politics/voter-id-election-la...
So what you're saying is that voter ID will result in some citizens not voting. Personally, that seems like a fair reason to oppose the measure.
Propose voter ID laws that include provisions to make sure that it is easy and free for eligible voters to obtain the necessary ID and Democrats won't object.
The poll linked from the CNN article demonstrates this: there's also overwhelming support for making voting easier, but voter IDs (without free and accessible ID services) will make it harder. And that's the crux of the common Democratic position: voter IDs are perfectly fine if we ensure access to IDs. But that needs to happen before or with any restrictions on voting, to prevent disenfranchisement.
But that's a topic the Democrats are very divided on, and the Republicans really don't want, so it's never gonna happen. The result is that I'm not against IDing voters in principle but am in practice, given the current political situation and lack of demonstrated urgency or need that might justify the down-sides of doing it absent universal national IDs.
I guess I just wonder how likely it would be to quash allegations of voter fraud. When we humans distrust a process, we seem to be able to come up with all sorts of wild stories to justify our fear and anger.