Am I reading your original message correctly when I assume your have made one connection -- and one only connection -- into the input side of your SB;
namely, from the set top cable box to your SB, using component cables from your cable box to your SB, and then component cables from your SB to your TV?
If I am correct, then the problem you describe will happen, since the cable box can only deliver one channel at a time.
To get what you want (i.e., stream one channel using the SB built-in tuner while the family watches on the local TV the pass-through channels from the cable box), you will need to put a signal splitter on the coax cable coming out of your wall, with one output from the splitter going to the coax input on the cable box and the other output from the splitter going to the coax input on the SB. Then the built-in tuner in the SB will interpret the signals coming from your wall. BUT NOTE, the SB tuner will not be able to interpret all of the channels that your cable box can interpret, but only those channels that (by law), your cable TV service provider is required to deliver unencrypted. In my situation, these include the local affiliates of the big networks (ABC, NBC, CBS), local community channels, plus some home shopping networks, religious networks, and foreign language channels.
In order for you to get your SB to stream every possible channel that your cable TV service provides you through the set top cable box, while at the same time allowing your family to watch the same full suite of channels (independently) on the local TV, then you would need to lease a second cable box, split the incoming signal from the wall (as described above), and put a cable box on both outputs from the splitter. In this situation, of course, the internal tuner of the SB becomes redundant.