If the aerial is connected directly to the input of the YouView box, and the aerial out is connected to your TV, then that is the correct arrangement to allow FreeView signals through.
The YouView box is then connected to the TVs HDMI input port, that allows all of the YouView SD, HD, and subscribed channels.
If that is how its connected, then BT may request a visit from one of their BT Vision engineers to confirm that there is nothing wrong with your signal, and you may have been unlucky with the two YouView boxes that were sent.
Do you subscribe to any of the extra channels, and do they work properly?
See what the TV experts say, they may request a visit before agreeing to cancel the TV part of your contract.
The YouView box, being a PVR, does need a stronger signal to ensure an error free digital extraction from the muxes before it is transferred to the hard disk. This also means that it will only accept a much narrow range of signal levels than a TV.
A TV can tolerate a lot of decoding errors before it becomes noticable on the picture, the same would apply to Now TV.
I would wait and see what BT suggest, as it would be far better to get it sorted. You need someone with a TV signal strength meter, so the actual signal level from all of the muxes can be measured.