A better question is why have a voting age at all, except maybe to establish the formal date when a child should physically cast his own vote. We don't (and shouldn't) abridge the rights of the mentally handicapped or knowingly ignorant adults, so what right do we have to stop children of any age from voting (or their parents voting in proxy for them) since those children have more at stake in the election than any 80 year old? They're the ones who will inherit our debts, infrastructure, education systems, etc.
Regarding the OP, your only 'not required to show who you are' because a statistically insignificant amount of times does anyone ever vote as someone else and the issue get raised. Voter fraud is a non-issue, while voter suppression and disenfranchisement is historically a very very serious and common issue, so we err on the side of letting people vote. The only reason to change that now is that the poors, young, and immobile old who might be restricted by these laws from voting more often vote Democratic than Republican.
Here in reality, things cost money. You can't simply give away everything to get elected while having fewer and fewer people pay taxes - at least if you believe in math. As a Mathocrat, I summarily reject your premise that any of the items you listed are inherently bad.
It's only "fewer and fewer people pay[ing] taxes" because fewer and fewer people have a vastly larger percentage of the total money and total income. I doubt there's anyone who would disagree that it would be fantastic if everyone in the country payed taxes, if that meant everyone was making enough money to be able to afford it. The way to broaden the number taxed is to broaden the median wealth in the country.