Phil,
There is no hard fast rule to motorcycle oil and why many like me still run dino.
Oil is complex and a lot of reasons really. If you have hung out in race circles, the enemy of engine wear and failure is shear. Motorcycle transmissions with wet clutch shread oil and why shear is so important. Benefits of syn? Viscosity stability (inverse temp to viscosity) and temperature management mostly. The very ingredients that mitigate viscosity loss at high temp are not the best for shear resilience. I run Rotella dino in the white jug. Bike runs and shifts beautifully on it. I have run Rotella blue in other bikes as well. Rotella T blue is NOT full syn. Its base stock is dino oil just like the white jug. It is a group III syn which allows the use of a syn name which incidentally is only allowed in the US. European markets do not honor this convention which is marketing driven. The reason that syn is not as attractive versus say in an automotive application is because the real benefit of syn is excellent lubricity. You don't want nth degree slippery oil in a wet clutch application. Triumph clutches are not known to be that robust to begin with.
So, you want good shear strength (shredding reduces viscosity and hence film layer of protection) and an oil that works well in a wet clutch application. This is why Rotella dino is so popular.
Here's the catch. The reason there isn't any resolution about this is its almost impossible to prove. Engines rarely fail now and dino oil is so good because of the technology of its additive pack that it easily rivals synthetic and even ester syns and the difference aside from price is miniscule. Nikasil cylinders are almost impossible to wear out on any oil as well.
Most important thing to do in your Triumph? Run the proper viscosity relative to ambient temperature outside and if you run the bike hard in particular which many of us like to, then change the oil frequently. Racers regardless of running dino or syn change the oil after EVERY race because it shears to about 1/2 its rated viscosity. That is the best advice. The type of oil is almost incidental and highly debatable as the differences are very slight.
Hope that helps.
PS: Btw, I am not denigrating synthetic oil. FWIW most motorcycle manufacturers spec semi syn or Group III oil for their bikes which is mostly comprised of dyno oil. In the case of Triumph they spec Mobil 1 4T which is hard to fault. The point is, it is very expensive to formulate fully synthetic oil and they pass this cost to their consumers for little if any value added. I have a gallon of it on deck in fact for my next change but no preconception that it is any better for my bike than Rotella dino.