You're correct, of course, modal verbs take a bare infinitive after them, not an infinitive with "to" – e.g. "you must arrive before 7.00pm" not "you must to arrive...".
However, there are, as you no doubt know, some verbs that can be used as either modal verbs or non-modal verbs. An example is "need" – it is used as a modal verb in "you need not worry about it" and as a non-modal verb in "you don't need to worry about it". Even in ordinary positive statements, it can often, but not always, be used either way: "you need worry about it only if you've made a mistake" and "you need to worry about it..." are both correct – but it wouldn't be fully idiomatic to say "you need arrive at 7.00pm"; most English speakers would say "you need to arrive at 7.00pm". The verb "dare" is another one that is used both as a modal verb and as a non-modal verb.
I am sure, however, that anyone who used a modal verb with an infinitive with "to" when that is not grammatically possible would not be able to explain it away by saying that the Home Office gets it wrong in some of its correspondence.
As to the usefulness of the "life" test, it is of course difficult to reduce a knowledge of "life in the UK" to a series of multiple-choice questions. It is ironic that so many British people would be unable to pass the test, although I suppose that most of these people would pass it if they studied for for it. However, that raises again the question of how useful the test really is: after all, I could no doubt study successfully to pass a test on "life in Papua–New Guinea" but my passing such a test in such a way wouldn't indicate that I know much about real life in that country.
I suppose the test was brought in by the previous government at least in part as a sort-of "knee-jerk" reaction to questions that were raised after the prominent terrorist attacks of the early years of the 2000s and a sudden fear (expressed with suitable outrage in parts of the popular press and media, for example) that people were being naturalised as British citizens without being integrated into British life. As with most "knee-jerk" reactions, the rationales for doing it were probably not very well thought through, to be honest.