Crazy question: possible to read values being sent to an LCD in a circuit?

Of course you can tap into any digital line,

My point is that LCD driving lines are not digital logic, but mulit level signals.
Also there are several different types of LCD, as explained.

You're not going to get that by decoding a $20 tuner either

Wouldn't be so sure, don't forget that you pay for software once only, with hardware you pay every time.

Detecting a note can be difficult because of the harmonic content, notice that tuners do not always give the octave but just the note. It is a mistake to think that the fundamental is always the largest harmonic or the lowest harmonic.