Yes - this ^^^^
I've never used Super 'Phatic (it sounds great!) but you can certainly do the same thing with extra-low-viscosity CA glue (number of makers/suppliers but it is easy to get hold of and is a specialist 'super glue' that is much, much thinner than the standard stuff )
As @Stub Mandrel says, capillary action sucks the glue deep into the joint. Masking tape right up to the edge is essential with CA glue, though, and after peeling it off, probably would need scraping with a razor blade to clean it up - and this might be where the super-phatic may well have the edge if it is water-based.
If you're fast with your fingers, you could even squirt the glue into the crack while the string tension is on, maximising the opening, then immediately release the tension when the glue's in there and, as @Stub Mandrel says, pop the pre-prepared clamps on.