The problem is that it proceeds from a false assumption, certainly in England and Wales; and without traffic figures it's very difficult to say where "had traffic taken away", as it becomes guesswork at that point. Yes, an all-purpose road that runs parallel to a motorway will obviously have traffic taken away from it; but motorways were designed to use mostly new corridors and to take traffic from a range of all-purpose roads. This is made clear by reading documentation such as
The London - Birmingham Motorway: Traffic and Economics from 1960 regarding the opening of the first section of the M1, plus M10 and M45.
In your comment that you've put on the
M5, you've quoted the A38, but you've also quoted the A46 as "sections were detrunked or made non-primary following the building of the M5", but which bits do you mean? A quick look at the Route Planning Maps on SABRE Maps will show that sections of the A46 were both Principal and non-primary prior to the opening of the M5, and my collection of early Landrangers indicate the same status after the opening of the M5, so the statement is inaccurate, or at least, unhelpful. Then it walks about the A429, a road that never gets particularly near the M5 with a speculative "may have lost traffic"; with no basis given for the claim.
The fact is that the M5 (when looked at holistically as part of the motorway network) will have relieved large parts of other roads in the vague vicinity, from the A449 and A435, to the A370 and A40, a huge number of roads in Bristol, Cheltenham, Gloucester, Worcester, the West Midlands conurbation etc etc etc, but without any evidence, claiming individual roads doesn't really tell anyone very much.