Didn't say it didn't happen, it did (frequently), but it was always unofficial and against the rules. It was prohibited by regulations and if anything had ever gone wrong a CM would have followed.
In the 70/80s BA used to fly round trip charity flights for kids on weekends around Xmas and ask to be intercepted, and we'd oblige. "No Santa on his sleigh kids, but if you look out the left hand window you'll see an RAF F4..."
The Norwich Flier got intercepted regularily, as stated, as did puddle jumpers for crossing the North Sea in the Great yarmouth area. WT had to man the CAP down there even though there were never any targets. So, if you saw a low, slow contact it was, "intercept with caution" (It'a civvie, don't scare him). As I recall one intercept, where the crew misjudged their closing speed, the call was, "low, slow, yellow, cesssnnnnnaaa" as they pulled about 9G to miss him.
You'd also "intercept" airways traffic by coordinating and maintaining a 1000ft below, at least when the aircraft was squawking Mode C. I recall one such intercept with a Shar.
SHar: "Visual"
Controller: "Roger, should be a SAS DC9".
SHar: "No, this is a XXX 737, and there's a woman in a yellow dress in the last row waving at me."
The most flak I ever saw was when the LU QRA was holding north of Saxa (usual Bear Fs that didn't penetrate) and was intercepting the atlantic traffic to pass the time. (At the time Saxa parked the labels on RAPDS and the controller and crew did intercepts whilst the DC in the Bunker, and any 2*s in their offices were unaware) When he got on the ground the bright spark of a Nav "claimed" a Korean Air 747 in the INTSUM to SACEUR. This was a couple of months after the Russians had shot one down. What a fun day that turned out to be**....
So:
Rule 1: Intercepts on Civil aircraft are prohibited.
Rule 2: If I ever intercepted one, I didn't know it was a civilian.**
Rule 3. If I did know it was a civilian, my aircraft never got closer than +/- 1000/2000 ft.
Rule 4: I deny everything.
(**The explanation given and, and I don't know how anyone up the ladder kept a straight face, was that the 747 was off the expected flight plan track and they thought it might be one of the Bears...)