Regarding the idea of a single remote control that works both the TV and a Sat box. Yes, that's possible. Your Logik TV should have something called HDMI CEC (HDMI control). Does your Grundig Freesat box connect to the TV via HDMI? If so, switching on HDMI control for both the box and the TV will give you the Grundig remote as the main controller. Switching on the box will turn on the TV and set it to the correct HDMI input. The volume buttons on the Grundig remote should operate the TV, but do check the manual as some boxes require the handset be programmed first. Switching off the Grundig box will turn off the TV.
HDMI control is a bit hit-or-miss with Humax boxes. You might want to stick to the Freesat 4K boxes from Argos that
@foxhole suggested. They'll also work fine with a 1080p TVs.
I put together the image below for you. This might not be for your TV, but it shows that HDMI CEC is a feature of some Logik TVs. .
Regarding the inability to receive BBC-1, there could be a few reasons for this.
* the box needs 2 sat cables attached, but you have only one
* there's a problem with the LNB (the lump at the end of the satellite dish arm where the cable(s) connect)
* a fault with the Grundig box
* some other undiagnosed issue relating to info you haven't yet supplied
The broadcast channels for satellite for the type of receiver that the Grundig is are split between four groups based on a combination of frequency (high or low) and polarisation (vertical or horizontal). The receiver switches between the four groups depending on the channel being received. ITV3 is the low frequency horizontal group.
Since BBC One is regional, there are multiple versions of BBC One depending on the TV region it's intended for. The regional variations are split across the horizontal and the vertical polarisations of the low frequency group. If you're able to get ITV 3, but you can't get BBC One, it would suggest that you live in a TV region where BBC One is in the low frequency vertical group. (see the graphic I made for you below)
Your reception problems could be an issue with the LNB, or there's a cabling issue which means the receiver is only seeing the horizontal signals.
If you're up to doing a bit of your own investigation work, you can use the channel and transponder listing here to work out if the issue is related to the polarisation. (FYI - high or low frequency should make no difference unless your box has a fault where it can't emit the 22kHz tone for high/low switching.)
https://en.satexpat.com/tv/uk/freesat/
Work your way through the list and write down the channels you can't get. If it's an LNB fault then a pattern should emerge that all vertical channels are affected.
