Well, you're going in the right direction with breaking down sentence into pieces.
As someone who's non-native and can somewhat understand JP (which ironically started with me finding 2 different translations of the same term, went to the source material and compare letters with similar thing until I sorta reverse-engineer the whole katakana...it was pokemon names btw) I'd say having the foundations down (sentence structure, different verb tenses) will get you quite far especially if you double check the material for context.
Yeah, figuring out the structure seems to be the key - without that it feels like the original is just a soup of concepts (the nouns and base verbs/adverbs/etc), that are just thrown together and you're expected to figure shit out somehow. Getting the main particles at least roughly in your head gives you the core relationships between the concepts, figuring out the verb forms gives you tenses and so forth, and then it's less of a soup of conceptual entrails you have to try and divine meaning from, it's kind of a conceptual sentence . . . Of course, then you have to try and figure out how to express that in English . . . And honestly, one of the nicest thing about the MTL is that it gives you a starting point there which you can then fix up, rather than having to come up with everything from scratch.
Trying to figure all this out is making me regret never learning any formal grammar - it was never necessary for me to understand English (and that lack of linguistic education hasn't stopped me from being very good at writing), it's only when I'm trying to understand things in a different language that I'm seeing how useful it would be to have a way to formalise the structure of language properly.
Anyway, it's kind of fun, though as I said it's the slowest and most painful way imaginable to read an interesting story.