In addition what is needed to learn English, there are some patterns to be learnt too. The grammar books make a good deal out being able to correctly identify and decode the elements of a word to recognise its grammatical function in the language. What is not covered so well are the similar segments of words which may or may not be (directly) related to meaning. An alphabet is an alphabet, but we recognise and read syllables (or even syllable groups) not letters.
Look at these two sets of words:
- ἀνείλατο, ἀνετείλατο, ἀνηγγείλατο, ἀντανείλατο, ἀπεστείλατο, ἀπηγγείλατο, ἀφείλατο, διείλατο, διεστείλατο, διηγγείλατο, εἴλατο, ἐνετείλατο, ἐξανετείλατο, ἐξαπεστείλατο, ἐξείλατο, ἐξηγγείλατο, ἐπεκείλατο, ἐπεστείλατο, ἐπηγγείλατο, ἐπωκείλατο, ἐστείλατο, καθείλατο, κατεστείλατο, κατηγγείλατο, παρηγγείλατο, περιείλατο, προείλατο, προεπηγγείλατο, προκατηγγείλατο, συναπεστείλατο, συνεστείλατο, ὑπεστείλατο
- ἀνείλετο, ἀντανείλετο, ἀφείλετο, διείλετο, εἴλετο, ἐξείλετο, καθείλετο, περιείλετο, προείλετο, προσωφείλετο, ὠφείλετο
To get to the situation you are describing, what needs to be known here? Obviously (in this example) the two terminating trisyllabic units - είλατο and είλετο, and in addition to that there are the (usually) monosyllabic initial syllables - the full or ellided prepositions (for the most part). Both of those "elements" should be fully familiar - seen as one unit, like to be able to be glanced at and written down with your eyes shut, etc.
Morphology and pattern-recognition only slightly interact with each other. The gap between patterns in the words and analytical morphology is something that natural language processing can deal with. Restricting pattern recognition to the morphology of the language has initially higher success in letting students parse, but contains so much (redundant) information that everything gets slowed down. Larger recognisable chunks - that may break morphemes or include more than one piece of information, are workable.
The declensions of number-case endings and in particular the article are usually set for learning outside of their natural patterns. I mean, we are never going to see ὁ τὸν τοῦ τῷ in a text, nor εἷς, μία, ἓν just listed together like that (Ephesians 4:5 has nouns with them). What we see are patterns of endings within nominal phrases. In this phrase πάσας τὰς βασιλείας the most important thing is not the endings that give us the parsing, it is the πάσ- and the βασιλεί- which gives us meaning. The other part, the xxx
ας τὰς xxxxx
ας should be processed as pattern-recognition - a lower-order brain function, while the "meaning" of πάσ- and βασιλεί- can be processed in working memory. Being able to spot patterns in sentences sub-consciously would be the known in this case - which you are saying that students should become familiar and comfortable with, and the πάσ- and βασιλεί- are the new information.
What about the accusative? Is it remarkable? What level of conscious processing needs to go into that? It should be so fully expected, that it becomes unremarkable. Habituation should let us expect that δείκνυσιν should be expressed together with a dative (in this case αὐτῷ) and an accusative (which we are discussing). It is the same with prepositions, there is not surprise to see a genitive following ἀπό.That expectation is something that should be known by habit. That is what happens when the verb occurs before the nouns.
In the other case, when a verb is held back to the end, there are a finite (but much larger) number of situations that are possible if a phrase started with simply an αὐτῷ, for example, the number of possibilities gets less and less (both in terms of possible structures and possible patters within various units) as the sentence goes on. That narrowing of possibilities is another skill that needs to be acquired. The overall pattern is not set up for us to follow, it becomes apparent later on as we have gotten through it. There is a lot more higher order thinking to be done in these situations.
Taking that further, the inconsistent nunnation (of the accusative) that apparently accompanied the gradual loss (merging in form) of the dative sometimes seems to force the unambiguous element forward so that the one that could be construed either way when heard is clearly what it is, but that is more for analysis than teaching.
Reading with minimal analysis and maximal pattern recognition better suits children's learning styles.
Building speed is not so much about doing minute (small) things faster, but about chunking and ignoring the obvious (things that a learning process should make so obvious that they are felt that they can be ignored in the higher order processing). Cf. Speed to Read