IME, powered aerial boosters are a waste of time. You’re amplifying the signal but also amplifying the interference, and you can’t magically add signal data that isn’t there. I’ve had a couple and neither made a jot of difference to the signal quality. Internal splitters aren’t great either (the only reason I can think of for an amplifier is to use it as a spltter rather than instead of one).
Your theory certainly makes sense in terms of amplification immediately before the TV (although potentially if amplification has some filters on it – it COULD clean up some messy signals in some circumstances, although there is no reason for a TV not to have similar capability inside, except cost). However your assumption is that all the noise that arrives at the TV was present at the aerial in the same ratio. Is that really true? There must be noise introduced over the length of the cable. There is a loss every time there is a junction in the cable/connector etc. There is a loss when there is a splitter. Now you might be able to achieve the same quality of signal at the TV just by using better cables, better connectors etc, but that is not quite the same as saying they are a waste of time – depending how hard it is to rip out the cables/fittings a booster might be much quicker and easier to fit!
By removing the plug you are removing the fuse. If its hard wired into a lighting circuit that’s even worse isn’t it?
What does the fuse protect? Presumably only the wiring after the fuse. If the appliance (which probably draws 100mA at most) is supplied with a 13A fuse, and you connect it to a lighting circuit which can only provide 5A (with a suitable fuse/CB) what does the 13A fuse do? when can it blow? Its certainly unconventional to wire anything except lights and fans to lighting circuits but is it actually more dangerous?