Now that I think of it, Quirrel watched over Harry as he felled a bunch of trees in Precautionary Measures pt.2. And that involved partial Transfiguration.

So was Voldemort really unaware that Harry could Partially Transfigure Things? Or did he only underestimate what could be done with that?

From our perspective, yes, I agree that it is obvious, but from Voldemort's perspective at the time, Partial Transfiguration just isn't something that exists; there's no reason to have it as a prominent hypothesis to explain Harry's cutting down trees. It's much simpler to assume that Harry had just figured out a more efficient variant of the Severing Charm, or something similar. As Voldemort himself said in chapter 108:

...experience has taught me that such puzzles are far harder to solve when you do not already know the solution...

[edited]