Do we really need quantum mechanics to get to semiconductor physics?
It depends what level of understanding you're interested in. For example, are you simply willing to take as gospel that somehow electrons in solids have different masses than electrons in a vacuum? And that they can have different effective masses along different direction of travel? That they follow a Fermi-Dirac distribution? That band gaps exist? Etc.
If you're willing to accept all these things (and more) as true and not worry about why they're true, then quantum mechanics isn't really needed. You can get very far in life modeling devices with semi-classical techniques.
However, if you want to understand why all that weird stuff happens in solids, then yes, you need to know quantum mechanics.
Can you actually derive transistor behavior from QM directly?
It depends on the type of transistor. If you're talking about a TFET (or other tunneling devices, like RTDs and Zener diodes), then I challenge you to derive its behavior without quantum mechanics! However, if you're talking about most common transistors (BJTs, JFETs, MOSFETs, etc.), then deriving their behavior from quantum mechanics is a lot of work because the systems are messy and electrons don't "act" very quantum because of their short coherence time in a messy environment. However, the semi-classical physics used for most semiconductor devices does absolutely have a quantum underpinning. But there's a good reason it's typically not taught from first principles.
Anecdote: One time, I was sitting next to my advisor at a conference, and there was a presentation that basically boiled down to modeling a MOSFET using non-equilibrium greens functions (which is a fairly advanced method from quantum mechanics). During the presentation, my advisor whispered to me something along the lines of: "Why the heck are they using NEGF to model a fricking MOSFET?!?" In other words, just because you can use quantum mechanics to model transistors, doesn't mean you should. There are much simpler methods that are just as accurate (if not more accurate).