What I don't understand is what it should do for a TV channel to try making Katy sound bad when she performs on them? Wouldn't any TV company try to the best of their abilities to achieve a flawless programme rather, in their own interest? That French channel recently is perhaps just a distinctive example for them not always being capable of doing so, but I wouldn't look for some unfathomable sinister conspiracy plot where such a one wouldn't make any sense.
Yes, when people suggest that Katy couldn't sing live my question to them is usually if they refer to a short TV appearance they've seen as that doesn't mean they heard her singing live; it means they were watching TV, LOL.
I watched all the videos, too, and from a singing point of view I wouldn't know what to suggest what to 'work on'. Only Katy can know what she has to work on, and whatever those things are she is doing it already because she's a highly professional singing performer who knows her stuff, on the highest practical level of knowing it. I can only really see that TV shows who want her for brief one song appearances somehow find it hard to cope with her, but that's their problem, not Katy's, to sort out.