Is there a way to build broad knowledge for sight reading pieces, or is it merely a matter of sufficiently immersing oneself by just practicing?
To me, being competent at sight-reading comes from immersive practice. But sight-reading itself helps you build broad knowledge, because it gives you the ability to sample lots and lots of pieces of music—not just the sounds, but the physicality.
And uniting mind and body in sight reading is an excellent complement to just listening to other people’s recordings as you explore the world of classical music. For example, let’s say you were capable of stumbling through a Beethoven sonata at low speeds via sight-reading. You might be able to play (badly) 1-2 pages of the sonata in the time it would take you to listen to it.
One way to really get familiar with a wide swath of the classical music literature in, say, an hour a day, might be to listen to 2 sonatas (each ~15 minutes), and sight-read two other sonatas for ~15 minutes. Try to space out your listenings and your sight-readings of the same piece. For example, listen to them in order from first to last, but sight read them in reverse order so that most of the listenings/playthroughs are spaced out. This capitalizes on the spacing effect.
In this way, you would be able to expose yourself to all 32 Beethoven sonatas in a couple weeks. I bet you’d have a much better memory of how they go if you did the sight reading + listening combo, rather than doubling the amount of listening and doing no sight reading.