While oils have changed the engineering tolerances that your motorcycle was manufactured to has not. Triumph specified the following: Engine oil- SAE 40, Transmission- SAE 50 or 80W90 gear oil (same viscosity) and primary chain case- SAE 20.
That being said, please remember that your bike uses flat tappets and oils high in ZDDP are best. Normally only multi-grades rather than mono grade oil is available at your local CDN Tire. I ran 15w40 Shell "Rotella T"in the engine of my TRW's as it at one time contained a lot of ZDDP, but I think it is now mostly gone from that formula. ZDDP is a wear reducing element that has been steadily reduced in oils. The Japanese have come up with a motorcycle standard for oil and it is very good, please see attached list of qualified oils.
http://www.jalos.or.jp/onfile/pdf/4T_EV_LIST.pdf
I would
NOT suggest that you use synthetic in your primary chaincase as the friction discs of the clutch are designed for less "slippery" oil and will chatter rather than smoothly engage. I also found them hard to free-up in the morning. This is especially true of old brit bikes where the repo parts are made offshore to emulate the originals and rarely include the newest friction materials.
Its going to leak, get over it. Carefully replacing the primary chaincase gaskets with small amounts of modern sealants helps a lot but there is little you can do when the breather is on the front of the crankcase and the engine has a vertical split that needs sealing. I notice it is already leaking in the back of the truck, just marking its spot.