Regarding home team in ET...

The home team in the second leg either earned the right (finished first in group for round of 16) or "won" the draw (quarters/semis) to host the second leg. I think having ET at home is a justifiable reward for earning/winning the spot to host the second leg. Yes, a draw is random, but I think winning should be rewarded. With away goals counting in ET, the "losing" team is rewarded.

I do like your concept of having to win outright in this scenario. That actually means more to me than my convoluted logic, so I'll have to remember it if this ever comes up again.