Christiaan,
I can see where you're coming from, and at face value it's kinda compelling, but I believe your simplifying assumptions are flat wrong, hence the problem - you've stated a couple of times you're ignoring '2nd order effects of thrust vector etc', yet you do pay attention to the 2nd order effect of the tilting lift vector. That's not cricket - if the 2nd order effect of 1 is significant, so is the other.
Staying well away from the vertical, I think/hope we all agree that in level flight, T=D, L=W (and we'll leave it at that).
Rotate to a 10degree climb, stabilise so we're unaccelerated.
Now, you rightly point out that the W vector is unchanged (I'm trying to stay in your earth parallel frame of reference), and the L vector is tilted slightly back. But the T vector is also tilted slightly up, and the D vector down; sure, it's small, but so is the backward tilt of L. Don't ignore it
So we now get that (vertical component of L)+(vertical component of T)-(vertical component of D)=W Nothing too radical so far.
Now the question is, 'does the component of T' make up for the loss of L, and vertical component of D?
If we climb at the same speed as we were in level flight, T must increase - we all know that intuitively, but the maths is harder in your frame of reference:
(H component of D)+(H component of L) = (H component of T).
I don't have any numbers, but I'd argue that that increase in T is what makes the (vertical component of T) term larger than the reduction in L due to the tilt of the lift vector.
As previously stated you're free to pick any frame of reference you choose, so long as all terms are treated equally to that frame - it's common practice in maths to choose a frame that simplifies the calculations:
It's much easier to visualise if you take the aircraft as the frame of reference, then the only thing that factors is W; Pitch the aircraft and whole frame of reference; T must increase to combat the component of W that is now acting rearward along the deck plane, and L must decrease because it is combating a reduced component of W.
And no, I don't know it all.. I'm working the maths as I go along. Quite happy to be proven wrong, and learn something.