@gwenandfraser You are correct, the ethernet ports on your Gateway HDPVR and portals are not active. The BlueCurve boxes also would not offer internet passthrough to your TV, though the BlueCurve TV box may offer access to content that you might be using your smart TV for currently, like Netflix, YouTube and Prime Video.
If you are having difficulty with the smart TV staying connected to your home wifi, that could be caused by wifi interference, or your TV being too far away from the modem. If you have a modem that broadcasts separate 2.4 and 5 GHz networks, which one is your TV connecting to? Do other devices have difficulty accessing wifi when in the same area as that TV?