You may have a couple issues. My '68 slide - no different than the rest - was glued in place and required carb cleaner to get it to free. Due to the lack of clearance, this is most easily done with the carb removed. The studs that hold the manifold make removal challenging if the "flexible" duct between the carb and the filter have stiffened. Remove that black duct first, then unbolt the carb and stick a plastic sandwich bag over the intake manifold.

You don't have to take the top set or cap off the carb to start feeing the slide from the barrel of the carb. Turn it upside down and spray it carefully into the intake and it'll flow around the slide. Do this a few times and don't let it dry in between applications. Now you should be able to unscrew the top set and liberate the slide. You'll note a groove in the slide that will allow you to work the cable out of the slide and free the lead slug at the end of the cable. Pull the cable out of the top set. If it doesn't come easy, don't force it. Take a pair of needle nose pliers and put a kink in the cable right at the lead slug at the end and it'll follow the contour of the top set out. Alternately, if the cable isn't serviceable, just cut the slug free. Study the routing of the cable if you're replacing it.

You should have all the slack you need. Your wiring harness has likely seen a lot of heat cycles and may be a little stiff and brittle so go easy. Get both screws out of the switch assembly and remove the brake lever just to get it out of the way. You don't need to get both clam shells of the switch assemblies too far separated in order to remove the throttle grip assembly. It simply twists out. Don't just start pulling them apart roughly. If you're working in a cold garage, a hair dryer will return some of the flexibility to the wiring harness. Be aware there's a pin in the bottom shell of the switch assembly that prevents inappropriate rotation. The pin can fall out complicating reassembly.

The slide will likely have a spectacular conglomeration of grease and dirt. This should be thoroughly removed and re-greased before reassembly. With the throttle slide assembly out of the way you should be able to clean the track on the switch assembly that will have matching filth and equally needs to be cleaned and re-greased.

The throttle cable is easily accessed at this point. I find it easiest to put the cable in going up from the carb, work it around the steering head, up through the gap at the bar mount, into the bar and out to the slot in the bar for the hinge throttle pipe - that part the cable actually attaches to that you should also clean and grease. Putting the switch assembly back together, if it doesn't allow the throttle slide assembly to fit into the track in the two switch assemblies, there's a good chance that locator pin has shifted. Connect the cable to the carb last after you've cleaned the carb. That kink in the cable can be removed once the slug has passed back though the topset before re-attaching to the slide.

Read up on carb cleaning. Your '68 is very sensitive to clogged passages in the idle jet - both the ones visible on the side AND the one that should pass from top to bottom. All of the interior passages deserve attention. I've seen missing gaskets on top sets, emulsion tubes, no float gasket (!?!) and all of it needs to be there and happy before you're happy.