I see your point, but I think the distinction you are missing is that votes and on-topicness (or lack thereof) are not one-and-the-same.
A question might get a bunch of downvotes because it's off-topic, but it also might get a bunch of downvotes because it was poorly worded, or because somebody didn't think it was well-researched. The same is true of ...
The Reversal badge is to give someone an incentive to answer a good question that got downvoted because others misunderstood it, or disagreed with the whole premise.
Discussion on SO about this:
Edits to Excerpts do not qualify you for the Research Assistant Badge.
Retagging a question does not qualify you for the Editor badge.
It looks like you may qualify for the Organizer badge though, maybe. The actual 'retagging' ability isn't actually available to you until 500 rep, but now that anyone can put in a suggested edit that doesn't matter as much. ...
Users at any level of reputation can always review suggested edits made to their own posts.
Looking at the first handful of single-reputation users on the badge list, it looks like all of them have approved a suggested edit to their own post. This counts as a review task and awards the badge.
"Days" are measured according to UTC, and consequently it's rather easy to "miss" a 24-hour window. Also, certain kinds of login-only activity don't count as accessing the site.
I can see the day you missed, so it's not a problem with the system, just a side-effect of how it works. Unfortunately you'll have to start over. :(
I've had this happen to me before while trying for this badge both here and on Stack Overflow - just opening the page on startup doesn't seem to count as visiting the site - you actually need to navigate to another page before it will count your visit. Just viewing your profile or a random question once a day (or 10 if you'd prefer) seems to be enough.
Self accepted answers do not count, authoritative answer here and here, where it also states that only answers older than 10 days (with no votes) are considered. Also wiki answers and deleted answers are not considered.
See also: List of all badges with full description.