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If the throttle cable snaps shut on its own I would leave it. That's a good thing, means everything is free. most times if a throttle only rolls back because you rolled it back, the problem lies in rust build up on the handle bar which causes the plastic sleeve to bind, on the handlebar. Removing the entire throttle assembly including the white sleeve from the handle bar, cleaning away all the rust. after all the rust is gone paint he handle bar with a good quality rust proofing paint but not too thick the sleeve needs to move freely on the handle bar. When dry lubricate with a good light rust proofing lubricant like fluid film, then reassemble One trick to lube the cables is to fill a small plastic bag with oil stick the free end of the cable ( whichever end you can get at with the least amount of work ) into the bag with the outer sheath just past the opening. tie a string tightly around the bag and sheath using a sack or millers knot (this is one time string is better than a zip tie) than hang it so oil can trickle from the bag down into the cable Leave over night or longer. The throttle cable divider is on the left side it is a flat foot ball shaped plastic box with 2 cables from the throttle, 2 cables to the carb and one cable from the cruise control diaphragm and it sits behind the upper fairing about 3-4 inches back of the upper corner of the rad. you will have to remove the entire upper fairing to get at it. Unless there is a real issue with it which is rare its not worth al the work to get at it. If I remember there is a a white multi grooved wheel where the cables attache too. This allows either the throttle cable or the cruise cable to operate the carb cable by way of the shared plastic wheel inside the box. Odds of anything inside that box going wrong is very slim.

I haven't had to do any carb work on my '86, but its' pretty tight in there. Maybe try a long locking hemostat? I haven't messed with the cables at all either, so I'm afraid I'm no help there.

I just use 3/4" P clamps from home depot and a couple 1/4"x20 bolts in the HD factory bracket. Making those hockey stick bracket was going to be a day project. Once I checked into it the P clamps were easy and cheap.

Drain the carbs. There are drain screws near the base of each carb and they will drain via the hoses sticking out the sides. It's messy, and because of that, I might not even drain them, but keep aware they're full of gas still if you don't. Remove the fuel-feed hose (from the fuel pump) in the top-center of the carb bank. It is the only hose with a clamp. Use a radiator pick to free it up (carefully) if it is stuck. Pay attention to how the hose is routed. Gas will spill. Remove the four vent hoses. Notice the path they follow so you can put it back together right. Remove the vacuum advance hose from the left front carb holder and bracket. Unscrew the choke cable retaining screw enough to free the cable. Slide the cable housing out of the retainer. Unwind the cable and slide the cable-end out with the cable aligned with the slot in the lever. Do not use pliers (you'll fray the wire.) Remove the pull side throttle cable. It is the outboard most cable. There's a clip that supports the two cables under the left fairing, just ahead of the adjuster. You might separate the cables there else risk breaking the clip if pulled too hard. Optionally, loosen (screw all the way on) the lock nut and then adjuster for the pull cable while carefully counting the turns so that accurate adjustment can be returned. This gives you a fraction of an inch extra play that I always seem to need. Pull the cable up and slide the cable through the slot in the inboard side of the cable retainer housing. Spinning the throttle wide open will help. You can now reach the cable-end and slide the free cable through the throttle-lever slot. This can be a bit frustrating. Maybe someone can help hold the throttle, etc. I also gently use the radiator hose pick to navigate the cable around. Again, do not do anything that would fray these cables else you will need to replace it immediately. Loosen the four hose clamps on the carb holders. Free each carb from its holder rubber carefully and individually. The application of penetrating oil may help. This is not a good place to have a vacuum leak by cracking, tearing, or puncturing a carb holder. Eventually the entire carb bank should begin to rock in the holders without a lot of restriction. Lift the carbs free from the holders and intake. Use a rocking motion and do not force. You may need to pry - gently - with a flat blade screw driver between the top of the carb holder and body of each carb to fully separate. Be careful not to damage the carb holders. Watch that cables and hoses on the right side of the bike are not interfering. Slide halfway out the left side of the frame while watching that you are not crimping throttle or choke cables. The engine intakes are partially exposed, be careful nothing foreign can drop in there at this point. Remove the push side throttle cable as it is now more accessible. Remove the carbs and set them aside. Pack a clean rag or paper towels in each carb holder to plug the intake, the crankcase breather pipe, and/or cover everything to keep it clean! Allow nothing down the engine intakes. You can have a quick peek inside at the valves though.