The selection of Freeview channels you can receive on your Youview (BT TV ) box will not be directly influenced by the speed or indeed reliabily of your broadband connection. The Freeview channels are transmitted on several different multiplexes ( typically depending on your transmitter , 3, 6 , 8 or 9 different multiplexes which are on different frequencies and may have different transmission power) . So if you are not able to tune in all the Freeview channels you expect there could be several possible causes . the first check there would be to to confirm when you bybass the Youview box and connect the aerial cable directly into your TV and retune the TV as to whether you receive the channels you list as missing.
Looking at the some of the other issues
AMC channel is available as a live channel in SD or HD on BT TV , there is not AMC 4K live channel in the BT TV guide (there is some 4K content from AMC - Snatch I believe that is accessed by the BT Player.
The BT Sport 4K channel 433 has recently had a name change to BT Sport Ultimate - if that no longer appears in the guide then maybe the broadband problem you mention has impacted your speed.
(probably a good idea to also do a soft restart- ie when not recording hold power button for about 8 seconds and allow the Youview box to restart)
Don't use the BT Tv app as a player on Windows so won't comment on your device allocation issue query.
