I think the issue is the wording and meaning of your question to Avanti.
I initially read "are Avanti trains not allowing" as "aren't Avanti trains allowing", which would be a valid question from someone who might not be aware that Birmingham to Euston includes Rugby to Euston. The response of "we are indeed" suggests that Miles at Avanti read it the same way.
I suspect what you meant was along the lines of "are Avanti refusing to accept LNR tickets between Rugby and Euston?".
Having said that, Miles from Avanti's response could have been clearer - "I can confirm that we are accepting LNR tickets between Birmingham/Rugby and Euston" or "Sorry, we are not accepting LNR tickets on our services".
So this may have been a miscommunication...
I asked for a cycle reservation on an Avanti service from Rugby to Euston but didn't receive it, instead received the response I reported here.
Avanti have now provided the requested reservation.
This came after having to fight my way up north earlier in the week, with boarding denied
by Chiltern for a few minutes at Marylebone with ticket acceptance in place "because your ticket's from Euston there's no ticket acceptance with Avanti" without looking in detail at my ticket. I said I was travelling with LNR and after the gate line staff checked with control came back to say that ticket acceptance was only in place with West Midlands Trains. So I then had to shout across that LNR is West Midlands Trains and the gates were opened remotely.
Then again at Birmingham New St booking office Avanti reluctant to provide a cycle reservation to Crewe when ticket acceptance was in place according to LNR. The 17:04 WMT service to Crewe had just been cancelled but the 17:07 Avanti service was on time.
LNR advice was ticket acceptance any reasonable route with Avanti to Crewe and I had to show a screenshot of this.
Surely when ticket acceptance is in place rail staff are updated ? Often not it seems.
I have a short fuse now