OTOH, I've been running Valvoline Synpower, which has relatively little zinc in it, for a long time in all my Triumphs; and I have yet to observe any damage that I attribute to oil. The engine in my TR3A was already worn when I got it (sitting in a field in Texas, in a TR3 that just happened to have an overdrive in it), had covered perhaps 150,000 miles since then, and was still running fine when it got wrecked. I did have some other problems, changed the bearings once or twice, replaced the head, but AFAIK those are still the original cam & lifters in there.
Best I can tell from the various literature, high levels of zinc are only needed in non-roller motors that have been fairly highly developed, with fast cam ramps and high rpm. We're talking engines that are so close to the edge that they routinely break through the oil film on the lifters & cam (which is where the zinc provides a benefit). A nearly-stock Triumph motor shouldn't have any trouble at all with modern oil, IMO. And since synthetic oils have a higher film strength, likely they don't need the zinc even in motors that do have trouble on "dinosaur juice".
But obviously I'm not going to guarantee that, and everyone seems to be recommending high zinc all the time because everyone else says the same thing. The FUD factor is very high (Fear, Uncertainty & Doubt) so no one is saying you don't need it, for fear they will be blamed for anything that goes wrong.
Including of course cams & tappets that aren't made correctly, which have been a problem almost as long as I've owned a Triumph; far longer than the flap over lack of zinc.