I know many Americans are skeptical of their government being responsible enough to handle their data, but they probably already have some data over their citizens (how else would the states figure out if the person registering to vote actually exists?)
This is incorrect because there is more than one government that is relevant to this question
Elections in these United States are run by the state governments. State governments likely do have all of the identify information you are talking about. The Federal government does not likely have all of this information, because it simply does not need to have it.
There are many people who value their privacy, and believe that the government should know only the minimum information about people which it actually needs to know. The federal government does not need to know my height, weight, physically identifying features or any other information typically on a national identity card; it does not impact the services provided to me by the federal government in any way (except international air travel). Therefore, the federal government simply does not have any legitimate reason to collect the information that a federal ID would require.
This attitude is likely less prevalent in Europe because the expectation of what services European national governments provide is different, but the principle of privacy in general is not itself that different.
So, why is there not more of a push to make national identity cards more available in the US? That is, not to make them mandatory, but just easily available...
There already is an available national identity card that is entirely optional: The U.S. Passport
While it may not meet the standard of "easily available" in your linked Harvard Law report (costs between $65 to $175 depending on whether you get the card, the book, or both), it's pretty much universally accepted for any identification purpose in the United States.
People don't feel a need for a national ID card
There is simply no activity other than international travel that Americans need a nationally-issued identity document for. State driver's licenses can be used for any routine identification purpose, and frequently are. People who do not drive can typically obtain an identity card from their state's issuing authority that serves all of the same purposes except operating a motor vehicle. The idea that the federal government would be somehow better at issuing identity documents than a state government, when it has no need for the information such documents would claim to validate, is extremely dubious.
Identifying voters in an election is not a valid reason for the federal government to issue identity cards because elections are run by state governments according to the Constitution
From Article I, Section 4 of the Constitution:
The Times, Places and Manner of holding Elections for Senators and Representatives, shall be prescribed in each State by the Legislature thereof; but the Congress may at any time by Law make or alter such Regulations, except as to the Places of chusing Senators.
The question of what that actually means in practice has been before the Supreme Court on a number of occasions, but is basically nearly anything except disadvantaging particular candidates for office as a class, so long as it's oriented around ensuring elections are fair, honest, and orderly.
People argue about whether or not ID requirements actually do that or not; I will refrain from opinining on that in this answer because I think it's vastly less relevant to why there is no federal ID document; recent discussion of "voter suppression" is something that has only emerged in the past decade, whereas debates about federal ID cards have been going on for a much longer time.
EDIT: There has been a slight disagreement in the comments with my claim that the federal government does not have sufficient information to identify people, on the basis that any information the states have is likely shared with the federal government. This is worth discussing briefly.
A working definition of identity for this context is that it is the set of unique traits and characteristics associated with a unique individual. To verify someone's identity, ideally you collect three types of information from a person and compare them to a database somewhere else:
- Something the subject knows
- Something the subject has
- Something the subject is
The last item there is very important with regard to identity documents; it's why ID cards list your picture, your height and weight, eye color, hair color, and other physical things about you. The document acts as something you have, and allows the person inspecting it to confirm that you have physical characteristics consistent with who you say you are, in addition to what you know about yourself like your date of birth.
Information on physical characteristics of American citizens is generally not gathered by the federal government, unless:
- You apply for a passport (because passports need this information)
- You are arrested for a crime (because law enforcement captures biometric information and typically shares it with the FBI)
- You undergo a civil background check that requires gathering this level of information. Employment in certain regulated industries is one such type of background check (e.g. like finance, aviation, law enforcement, the military, defense contracting, and even education). Another one would be the naturalization process.
The number of people that the federal government currently has this level of information on is approximately 139 million as of May 2021. The population of the United States was 328 million in 2019, so this is hardly a complete picture of the country.
"But what about people who pay income taxes?" The IRS knows that someone with your name and your Taxpayer Identification Number paid taxes, and on what sort of income. IRS cannot perform identity verification because it does not know anything else about you besides what is on your tax return, which does not even need to be filed by you personally.
"But what about people who use banks?" The information that banks collect is enough to satisfy federal government reporting requirements. It is not enough for the federal government to identify you at a later time.
"But don't you think states share this information with the Federal government?" Umm, no? We don't have a national ID database in the US, unless you get arrested or undergo a certain kind of criminal background check (again, see link 1).