But either way, someone getting started in the field has to at least learn to read it
At some point, definitely; just not before you understand the fundamental concepts they convey, and not while they are learning the very basics.
Do remember that this particular notation is very straightforward to explain in plain English, via
"if ... are all true, then ... is true", "
implies" (for ⇒), "
consequently" (for ⊦), "
formula" (for ⊥ and ⊤), and so on, although one must be very careful in the language, and must explain the differences and nuances (like the difference between "
then all of ... are true" and "
then at least one of ... is true").
I am not suggesting one invent yet another notation, just that explaining the basic logical system and these very dense inference trees in native human language, perhaps in parallel with the notation (if such dense notation is necessary), or better yet, in both human language and in other notation styles like Suppes-Lemmon and (augmented) Backus-Naur form, to convey the information more effectively; not limiting or selecting for students who concentrate on specific notation styles instead of the content and logic therein, but adding context and possible approaches for the students to fully understand instead of just memorizing the content. Essentially, that Spoken Human Languages :: Compiler Theory = Pinyin :: Standard Chinese.