I can see ADC being super hard to carry from if you aren't at a skill level much above where you're currently at, but obviously you can see pros carry at ADC all the time from lower MMR (which to them might be plat/low diamond). Carrying is not about 1v5'ing but picking smart fights you can win.
Your last game last night, Penalty, as Tristana, the main thing I could point out was that you were initiating with your jump and got stuck in the middle of enemies and then killed quite a few times because you wanted to kill enemy Sivir. As an ADC though, you typically want to stay on the outskirts and kill closest targets. You can use your jump to finish off a low health enemy knowing you'll get a quick reset but I watched you jump toward a full health Sivir and no reset was coming since you couldn't burst her.
We all make dumb decisions, especially when tired, but that was the main thing I saw which might have turned the game around in your favor if you could have killed other targets and stayed alive doing so.