I have asked a moderator to provide assistance, they will post an invite on this thread.
They are the only BT employees on this forum, and are based in the UK.
There is no reason why your service cannot be repaired. A moderator can take ownership of this, until its fixed. If the router is rebooting, then its probably faulty.
If the the external network is faulty, then it will still be faulty if you change provider. You can only cancel without penalty if BT agree in writing, that they cannot fix the problem.
You should get a reply tomorrow morning.
@Rossjohn123 welcome to the community and thanks for posting, I'm really sorry to see that you've been experiencing ongoing issues with your broadband connection. It really shouldn't take six engineer visits to resolve a fault and I appreciate you've now reached the end of your tether with the situation. The mod team will be happy to help with this if you send us over your details on the 'click here to contact the mods' link in my forum profile. You can find the link by clicking on my username.
Hi @Rossjohn123 thanks for sending in your details, sorry I missed you when I called. I have re-opened your complaint, and will deal with it until this fault is sorted, please don't worry.
I see a fault was sent back to our suppliers on Friday, so I will check it again tomorrow for updates as it looks like they found a fault on the network, which sounds encouraging.