"Only Xs are P" never means "All Xs are P", but rather "The only things that are P are Xs", which is equivalent to "All those that are P are Xs". However, "Only those Xs are P" does mean "All and only the Xs are P". This is because "those" somehow makes the phrase refer to the whole group of "Xs", and so none are left out when asserting that they are "P".
In this case, therefore, "Only those who ignore the facts are likely to be mistaken." means "Those who ignore the facts, and no one else, are likely to be mistaken." This is of course false in the real world, since there are people who do not ignore the facts but are mistaken anyway because they are not aware of the facts. But it is exactly what is stated by the English sentence. You then see that the argument is valid, but unsound. So you can say that this question is just a matter of understanding English, not logic.
Those who disbelieve my answer can check the Corpus of Historical American English for themselves. To prevent bias, I provide here the first 20 results for the search string ". Only those" in the past 50 years or so. Almost all of them provide unambiguous evidence for the semantics I claimed. The rest should also be interpreted in the same manner.
Only those who have been self-supporting for a year are eligible for relief.
Only those who have worked for their money ever acknowledge, in their deepest consciousness, the [...]
Only those people are going to be hard to know.
Only those West Berliners who have immediate relatives in the Communist part of the city may pass [...]
Only those forms fittest for their particular environment survived this process of natural selection, the rest [...]
Only those who work the miracle know what the hazards are, and they are not likely to [...]
Only those vibrational distortions of the molecule which give rise to a change of dipole moment interact [...]
Only those who've locked themselves out of society by barricading their doors with research books or [...]
Only those who have suffered ever come to have a heart that is wise.
Only those with gun permits would be allowed to buy ammunition.
Only those near the door knew at first there was a shooting.
Only those in the most extreme isolation, and these are atypical, fail to have an [...]
Only those suffering from that plague called liberalism can say that these restrictions put by a legitimate [...]
Only those who have denied their being yearn to play at it.
Only those who have been through the process can truly understand how far superior is an education [...]
Only those were safe who never loved and did not love. = Only those who never loved and did not love were safe.
Only those local organizations can do the job of combining neighborhood resources and talents with the financial [...]
Only those changes are held to constitute progress which directly or indirectly tend to heighten human happiness [...]
Only those in the cockpit and the first few rows of seats were spared.