Thanks for inviting me in on this. My specialism is hydraulics and the insides of clutches on Suzuki SV and VStrom thousands. No first hand experience of the Triumph triples but had some tinkering with cable clutches.
That picture of the extended lever on the clutch lifter arm and other chat on the forums does suggest a heavy feeling clutch is common on these Triumph triples.
I know much has been mentioned before but starting at the lever working our way towards the clutch its self, we need to minimise all the small things that might add up to make the clutch feel as bad as it is.
Hopefully the handle bar lever is standard length and distance from its pivot hole to cable end is standard to give original leverage (mechanical advantage).
Ensure the area where your fingers pull on the lever are as far from the pivot as can be. Some bikes end up with the handlebar clamp for the lever being too close to the grip (too far to the left). Often there will be a punch mark (round dent) in the handlebar finish. The join in the clamp should align with this and for maximum leverage the clamp end should be up to or just over the punch mark. If no marks then the clamp for the lever needs to be as far to the right along the bar without the ball of the lever fouling your little finger as you pull the lever to the handlebar.
Lever pivot is lubricated and lever is not fouling anything in the clamp?
Cable end is well lubricated and rotates a little in the end of the lever as its pulled and released.
Is the cable really OK. Cables that look and feel absolutely fine without damage and in good condition on the outside can hide a nasty little problem inside. Over the miles the cable that is already coping with a heavy to pull clutch and perhaps pulled over some tight bends against the fame will wear heavily at one single point inside. Without any load they still feel fine but as that inner cable has always pulled tight over the exact same parts of the outer, it starts to cut into the outer (or eats through the nylon coating between inner and outer). The inner gets "gripped" deeper into the slot its worn only when under actual load. Not felt when pulling the lever or cable without the other end connected to the lifter arm at the bottom.
I know it hurts to risk upwards of 30 quid on an experiment to find out if its the cable or something else. Can the cable outer be rotated 180 degrees along its whole length? That is 180 in relation the frame and the cable inner. Slacken any brackets and turn the outer top and bottom at the same time (and in the same direction). Any cutting of inner into outer will be repositioned and the clutch lever pull might be improved significantly. Its seldom a cure long term but good indication a new cable and one of possibly heavier duty might be all that is needed.
Is the pivot where the bottom of the cable pulls on the clutch lifter arm lubricated and running free?
From here on its all going on inside the engine cover.
Is the clutch lifter arm easy to move in the other direction to when its pulling the clutch? This will be with (occasionally against) the little weak spring on its top. Disconnect the cable and push it in the opposite direction to which the cable pulls. With luck this eliminates the lifter arm binding in its two bearings in the clutch cover. These bearings are within the area where engine oil is thrown about but the top one that is immediately beneath the little oil/dust seal and return spring on the clutch lifter arm can still fill with water and corrode. The oil/dust seal can fail due to wear or more likely get damaged due to corrosion of the steel lifter arm while standing over a winter or two. Besides facing up to catch rain and road dirt its easily overwhelmed when pressure washing the side of the engine. If the lifter arms shaft is binding in the cover then the only real solution is to change the lifter arm shaft and housing. Cleaning it out and greasing the top bearing might help short term.
Next thing on the way down could be wear of the lifter arm shafts end and the part it pulls through the release bearing centre to disengage the clutch. The shafts and release bearing centre are sometimes simple flats or in other engines have gear teeth so work as a mini rack and pinions. Either way its all in oil but if Triumph skimped on the sizes of the loaded faces there could be wear that increases friction or more likely reduces the leverage effect (mechanical advantage).
I can't imagine a previous owner decided to change the clutch pressure springs for heavier ones. This will make the lever far too heavy and wear all the items above prematurely. It does happen though where someone burned their clutch out due to abuse and blamed the springs for being weak. After market clutch suppliers all too keen to supply "uprated springs" with their new plate packs to avoid a repeat burnout that they might be held responsible for.
And if all checks out well there are two possible remedies. Try the extended lever on the end of the lifter arm but be mindful of the comments from others that your clutch might not fully disengage due to reduced travel. That might be overcome if you can cope with a handlebar lever a generous distance from the bar (as for wide finger span) or thinner grip so the lever travels a slightly greater distance. The other is have a dabble with lighter pressure plate springs. If six springs its easy to try reduced rates for just three. Then try reduced rates for all six. Shock horror guess some think. Manufacturers do their best to avoid their products getting a bad name, even from the abusers because abusers don't know they are. Clutches get all sorts of abuse so manufacturers will often use heavier pressure springs to avoid burn outs due to abuse at unbelievably high power outputs. If a sensible owner is aware his springs have been downgraded and avoids abuse, then there is no reason why the clutch can't be used as normal and the benefit of a 30% lighter lever enjoyed for ever.
Experimenting is easy. Just the cover off and if left on the side stand no need to drain the oil. Might see even on the main or paddock stand the oil level in the sight window is below the lowest part of the clutch cover. Trying with just three weaker springs (about 50% of stock) evenly spaced means three original springs and their bolts are still in place so nothing falls apart on you.
Well someone did ask. Hope some made it this far