Let's work with these summarises of Einstein's argument and Bohr's response, the latter repeating the former's assumptions about relativity and gravity. The response notes$$\color{red}{\Delta E}\color{blue}{\Delta t}=\color{red}{c^2\Delta m}\color{blue}{c^{-2}gt\Delta q}=\color{orange}{gt\Delta m}\Delta q\ge\color{orange}{\Delta p}\Delta q,$$where the red quantities are equal by special relativity, the blue quantities are equal by general relativity, and the orange quantities are equal by the acceleration $g$ Galileo knew about.
Does this thought experiment prove QM implies the coloured parts? No.
In an "I can get less $\Delta E\Delta t$ than Heisenberg said" argument, the explanation brings in extraneous-to-QM physics (in this case, the coloured parts). A counterargument uses that extra physics, which isn't a consequence of QM in either argument's view. If Einstein's argument works, QM is incompatible with the physics cited; if it doesn't, they may be compatible, but one needn't imply the other. If a third physicist doesn't grant these other ideas, that doesn't contradict QM; it just means they can't use them to work out what happens in the experiment.