I think Fr. William G. Most's Latin by the Natural Method is overall a better Latin textbook than Lingua Latina per se Illustrata, especially if you intend to use Latin as a spoken language.
Both books are an attempt to create a Latin version of the Nature Method Institute's language textbooks. They're coming from the same direction, as it were, and both want to teach you Latin by showing you how to read extensively. It would be a mistake, though, to characterize either as a reading-only approach. Both books are structured around a grammatical progression, even if Ørberg chooses to explain that grammar in Latin versus Fr. Most's English explanations. In this regard, Ørberg seems to win out, since his book has more Latin text from which to read extensively.
But I don't think it's that simple. The first volume of Latin by the Natural Method certainly has less Latin than Familia Romana, but both books have a second volume. Roma Aeterna is less a graded reader, and more a survey of Latin literature with a bent to historical works. The first few chapters are original Latin by Ørberg, but it quickly becomes lightly adapted Livy. By contrast, the second volume of Fr. Most's book is a meaty, extended narrative using the full range of grammar learned in the last chapter of the first volume at the same level of difficulty as that chapter. It's like if Ørberg had written another three-hundred page book that went between Familia Romana and Roma Aeterna, continuing the family's story at the level of the poetry chapter. Thus, I think comparing the two volumes of Fr. Most before he gives "real Latin" are about comparable to Ørberg's one volume.
But sheer volume is only one factor. Assuming you buy into the whole "compelling and comprehensible input" shebang, Fr. Most does both those adjectives better than Ørberg. Take this first sentence from the first chapter of Roma Aeterna:
Urbs Roma in ripa Tiberis sita est viginti milia passuum a mari.
Compare that to the first sentence of Fr. Most's second volume:
Ultimo anno, audivimus multas narrationes ex historia Romana, et legimus in vetere testamento.
Keep in mind that when those two sentences are read, their respective authors have both taught all the major points of Latin syntax and morphology. One of them is using much more comprehensible language, though. Ørberg will only ramp up the difficulty from there. Fr. Most has two hundred or so pages of that same difficulty for you.
Though both textbooks are ultimately based on a grammatical progression, Fr. Most chooses a more natural one than Ørberg, who still feels beholden to the order of concepts as they are listed in a grammar. Ørberg teaches all five cases, then the verbs starting with the imperfects, then the perfects, then the moods in the same order as the tenses, sprinkling bits of syntax here and there. It works, but it's clearly arranged on the basis of a grammar book.
Fr. Most starts you with the accusative case (in all three main declensions, not just the first and second) and the perfect tense. Then the ablative. Throw in the ablative absolute in the first fifteen chapters, and get the imperfect subjunctive in early. From there, he bounces around the grammar book in a somewhat free-flowing way. There's a logic to it, though. The perfect is more predictable than the present. If you know the perfect, then the imperfect subjunctive in purpose clauses becomes an easy way to constantly expose the reader to the infinitive form, which will be useful when they need the less-predictable present tense. If you know the ablative and some participles, why not toss in the ablative absolute? It gives you opportunity to constantly drill the passive participle. And if you know the ablative absolute, learning cum clauses with the pluperfect subjunctive is a breeze.
The whole book follows that kind of logic. Like unlocking a puzzlebox, rather than following the instructions to build one as Ørberg is doing.
It's a point of tastes, and yours may differ from mine, but Fr. Most's narratives are more inherently interesting than Ørberg's family, whose domesticity is refreshing when you've read volumes of Roman propaganda. When, in other words, you're a Latin teacher, and hardly anyone else. Fr. Most covers the history of the Roman Republic, paraphrasing Livy and focusing on characters. He switches halfway through to narrating the major episodes of the Old Testament. Along the way, he sprinkles in the Epic of Gilgamesh, a a few dialogues on Egyptian religion, several parrot jokes, his own opinions on the interpretation of scripture, and this odd sense of humor consisting mostly of screwball overextensions of logic. It's fun reading, besides being simple.
Fr. Most's book is also, in my subjective opinion, more useful to someone who wants to learn to speak Latin. Ørberg wrote his textbook to teach you to read the classics. I've been made to understand he was somewhat surprised to see it used in "active Latin" classrooms. Fr. Most, on the other hand, wrote his book for teaching Catholic seminarians who would need not only to read, but also to listen to, speak, and write Latin, and do it on a short timeline. My Latin teacher in his seminary days needed to be able to understand spoken Latin and be able to do his exams in Latin before his third year, because classes stopped being in English at that point. Fr. Most was their text of choice.
And with good reason. The complexity of sentences in Latin by the Natural Method matches the complexity of a skilled speaker talking to a beginner, not that of literature. Imitating Orberg, you'll learn to sound like a book. Fr. Most is dictating his spoken words to you, more or less, and you'll pick up from him how people speak. Furthermore, and this is an almost utterly unknown point, Fr. Most's second volume includes a "how do you say" section, where he covers a wide range of tricky turns of phrase and different ways to express them in Latin. It's tremendously useful, if I do say so myself.
Latin by the Natural Method has a few shortcomings. It tends slightly Ecclesiastical, introducing you to dixit quod before the accusative and infinitive. But it does teach you that. It starts with English-like word order. But it explicitly teaches other word orders, making a point of saying "in this chapter, we'll be learning to read sentences with the object before the verb, with adjectives separated from their nouns, with 'word sandwiches', with prepositions between their object and its adjective," and so on. It lacks vowel length marks. It doesn't have examples of as many fine little points of style and grammar than Ørberg skillfully sneaks into his readings.
But you can get all those things elsewhere. Like from Ørberg, for instance. Who will be a much more digestible experience if you already read simple Latin and understand the standard textbook range of morphology and syntax. There's no reason for this to be either-or. The point I want to make is that, pound for pound, I'd rather introduce you to Latin via Latin by the Natural Method than Familia Romana.