Honda-SOHC

Good news with progress. Potentially running more in equilibrium now and give better conditions for running rings in.

On that note, I'd not worry at using it up to about 7500rpm during this period. That's the torque peak and avoids "lugging" it to risk heat building exces.

My view is that the components generally don't need running in and associated care, just consideration for the rings/bores in bringing those in. That operation is related to linear speed, of the pistons, ultimately to "hone" the rings with the bore cross hatching.  These motor are so short stroke its unlikely you impinge on ideal linear speed for that action by running to the above rpm. Light throttle and let it spin with casual ease to treat it well 

Ah ha, different scenario then 

As Sesman notes, double check camshaft timing to make sure there's no mistake there. They'll start snd run at a tooth out, but not correctly  ------ and certainly done before, to much consternation. 

Carb sync, bench sync is absolutely fine to both get running and avoid faults tying you in knots, don't go near it with any gauges while diagnosis is happening.  It compensates for nothing in the engine and there's only one correct position for them, which decent method in bench sync will achieve. 

See if you can get definitive answer on cam position first, then judge where to go.

As above, can be quantified or could be checked against a "gauge" with the spark plugs being 1.25mm pitch, take the bleed nipple and drop threads into each other to get confirmation if it matches or differs by plus or minus to indicate which pitch you have.

Hopefully you'll get to the bottom of it with the coil rebuild, then verify its firing properly on all 4 cylinder.

Running on three will cause it to be a heat risk, as you've found. They've only about 24 ft/lb of torque @ 7500rpm when healthy, so starting with approx 18 lb and running below peak torque will likely bring about 15 lb at a, licked finger in the wind, estimate.

Trying to pull uphill like that will severely compromise its output, ultimately to cause lean running through that set of circumstances.  Essentially, the crank rpm can not "follow" throttle opening as there's insufficient torque to accelerate the bike and rider. That causes you to open the throttle further, which drops the vacuum affecting jets, to then run permanently lean (if the crank rpm  never catches the throttle slide position) up gradient being the biggest risk, as it can never catch up.

"Lean" running, in and of itself doesn't necessarily increase heat as a cause. What happens as the combustion  chamber reaches its ability to burn fuel as it approaches the 14.7:1 ideal (stoichiometric) the propagation of flame front outward from the spark plug begins to break down such that the burn phase doesn't spread out to far reaches of tbe periphery of piston/chamber. This results in unburnt "gas" mixture in a ring around that periphery, which then explodes rather than burns as piston comes to tdc. Its this uncontrolled event that gives a shock wave that breaks the boundary layer protecting the piston crown from exposure to high heat and subsequently raises the overall temperature if left unchecked.  When more extreme, we can hear this as "pinking" but it starts in virtually inaudible levels before we become aware of it in more severe form (we're isolated from source & with helmet sound insulation too) but it happens and significantly shifts the temperature accumulation, that also being relative to the ambient air trying to cool tbe motor and it's capacity if moving more slowly.  Detonation is the process, as there's loss of control thats fundamental to how the fuel air mix is supposed to combust, rather than explode. Essentially it has the effect of advancing the timing to detriment of the engine's desirable running condition.

There's another phase going on from this as too much of the above will likely bring pre-ignition, which is usually catastrophic, and in short time frame too  this will definitely put a hole in tbe piston for you!

Seems like you've got away with it in recent description, but quite close to serious damage though.

They are huge when you get up close, aren't they 

The classic "peanut" triumph tank shape would appear to be driver of sales in customer ideals I'd guess. Away from their more sport bike focus it looks to be significant in it's attraction to the brand.

Note that the underline curved arc on Triumph logo finishes early, rather than the original that loops under and past the "H" which I think comes from ownership of original script.

I didn't know that, but a "kind" observer pointed it out to me on a T-shirt I was wearing  with comment of "that's not a real one" 

Not my own bike, a friend's example when they were more contemporary, reached 120,000miles without rebuild. It was, by that time more compromised as it sounded the traditional rattle/tinkle signature they give out when the camchain "horseshoe" gets stuck from gummed and baked oil residue.

From recall, starter motor replaced but no other notable problems. Serviced regularly, but not with such absolute diligence that may have given more life, potentially.  It didn't fail in the motor, more put out of use by replacement with a newer bike alternative.  I didn't get to see inside it, but bores / compression etc appeared to be in order from how it started and ran.

Certainly, I feel that oils used then were more prone to gumming effects, with black sludge typical in many an engine in that period, along with typical (of the time) British outlook of extreme suspicion in what were relatively low mileage interval for oil changes. Thsts also accompanied by an economic leaning of "how much is that going to cost" generally echoed around dealer service desk conversation 

170,000 miles is certainly a pretty high watermark for the one mentioned in thread title.

Price isn't bad, but depends if you want to buy it.

Seals, try listing them, plus the camchain tensioner and primary drive rubber as those are most likely a controlling factor for economics of running such an engine.

It's a strip, inspect & replace the vulnerable parts to give it decent chance of fulfillment in happy future miles.

If the rest is good, it will,ultimately be worthwhile for somebody looking for original model in that condition. 

Some buyers will be losing that whole amount in depreciation on a new car, if not careful. This bike will likely keep its cost over coming years with, all but, nil depreciation.

The A-preciation will be in the happy miles spent riding it.

A couple of things to try in helping analysis and find out effect.

1) spray E-Z-start into airbox prior to cranking and see if there's any change to how it initially fires when cold.

2) cover airbox entry (thick rag or similar) and crank with kill switch off, just for a few 5 second bursts, then remove rag, switch on and try to start. Again, to observe if there's any immediate change in willingness to fire when cold etc.