If you chose to use BTTV and instead of using an antenna input to watch Freeview channels use the IP option , then obviously if your broadband is down your TV watching is affected.
If you are having problems with your broadband, but various engineer checks state they can find no issue , you may need to be more specific about what the actual problem is,
Are you on FTTP ( fibre to the premises ) or FTTC ( fibre to the cabinet ) if FTTP , when you lose your broadband what lights are lit on the ONT ( does the LOS light come on ) , if FTTC , what colour light is lit on the BT Hub when you lose broadband, does it stay blue or for example go green or orange ?, are you only connecting using WiFi , or do you have wired devices are are they also affected, is it certain times of the day these drop out occur ( things like faulty central heating devices switching on and off can cause interference than causes FTTC /ADSL broadband to drop out ) there are so many variables , to have a chance of diagnosing what the issue could be , much more information is needed .
Perhaps you can provide more details , what you have checked yourself and what have the Openreach engineers told you , an engineer stating ‘Yep , there is a problem but there is nothing they can do’ seems odd , even if that engineer had limitations on what they are able to check themselves , they should pass the problem to others OR or BT engineers that do have the necessary ‘skills’ to resolve whatever the problem is.
