From what I've read, I don't believe that's an accurate characterisation. But sources seem slightly messy and contradictory.
Engine 3 failed during rotation.
The crew misinterpreted the loss of water injection on the failed engine 3 as meaning they had run out of water, and cut the water injection to the other three engines, significantly degrading thrust. Thrust was also not set to go-around thrust levels after the engine failure.
(ASN says the water injection was erroneously turned off before engine 3 failed and this caused the engine to overheat and fail, but this is inconsistent with the Boeing letter and my understanding of the physics of water injection)
Remaining thrust was barely sufficient to keep them straight, level, and not-stalled; there was no significant climb or acceleration until the thrust levers were advanced further. It looks like the remaining injection water was dumped rather than injected?
I would very much like to see the original FDR traces referred to in the letters but have not found a source.
TL;DR great airmanship from the pilots with thrust present; not so good from the F/E engine management.