The difficulty you are having with squaring the burden to getting a photo ID with the fact that "...every state seems to have an option to provide proof of eligibility that doesn't require any money" is because it is not accurate to say that every state has an alternative to the photo ID.
Looking more closely at the NCSL source that you quoted in the question, you can see that Georgia, Tennessee, and Kansas fall into the category of "Strict Photo" and will accept only a photo ID before counting someone's vote. It is true that each state will allow the voter to cast a provisional ballot if they did not have the ID with them at the time of voting, but they each require the voter to show proof of the photo ID within several days of the election in order for the ballot to be counted. From the NCSL's analysis:
A voter who is unable or refuses to provide current and valid
identification may vote a provisional ballot.
To have his or her ballot counted, the voter must provide a valid form
of identification to the county election officer in person or provide
a copy by mail or electronic means before the meeting of the county
board of canvassers.
If a voter is unable to present the proper evidence of identification,
then the voter will be entitled to vote by provisional ballot in the
manner detailed in the bill. The provisional ballot will only be
counted if the voter provides the proper evidence of identification to
the administrator of elections or the administrator's designee by the
close of business on the second business day after the election.
If you show up to vote and you do not have one of the acceptable forms
of photo identification, you can still vote a provisional ballot. You
will have up to three days after the election to present appropriate
photo identification at your county registrar's office in order for
your provisional ballot to be counted.
With this in mind, looking at the real costs of obtaining a photo ID, we can see costs easily reach or pass the $25 threshold you highlighted. If you are a voter without a valid photo ID in one of these states the first challenge would be getting to the state license office to request one. A report from the Brennen Center and summarized in this Washington Post article found in an analysis of the 10 most restrictive states that:
nearly half a million eligible voters in the 10 states do not have
access to a vehicle and live more than 10 miles from their nearest
state ID-issuing office.
In addition to finding transportation to the office, they would need to schedule a time when they are not at work or other obligation AND when the office was open. The same report found that:
Rural areas in Alabama, Georgia, Mississippi, and Texas are served by
part-time ID offices. And in an extreme example, the researchers found
the office in Sauk City, Wis., is open only on the fifth Wednesday of
any month. That would limit the office to being open just four days
Finally, there is the cost of the documentation to secure the ID itself.
The report said birth certificates can cost between $8 and $25.
Marriage licenses, required in some states for married women whose
birth certificates include a maiden name, can cost between $8 and $20.