Nothing against 'dog's idea of replacing the core, if you want is an end-product which is identical to the original -- and as vulnerable to rot as it was the day it left the factory, with a somewhat cosmetically-flawed top skin. Personally, I like my repairs to be at least a little better than the original work, if at all reasonable.
However, regarding "re-coring the area properly", my considered opinion is that there is no such thing as a "proper" balsa core, ever. Coring is a weight-saving and economy-minded stiffening technique. Balsa coring, with its inherent rot risks -- from even a minuscule leak -- is a cheap way to stiffen the laminate. Foam coring is a different story, of course, since foam doesn't rot like balsa.
Balsa cored decks are perfectly fine, if people took the care to seal any penetrations through the core material. Balsa wood doesn't rot if there is no water getting into it. It also has some characteristics that actually make it a better core material than foam in many cases. First, due to the end-grain nature of balsa used in core applications, it tends to keep any damage relatively localized for a fairly long time. A small leak into a foam core can often end up in large areas of delamination, due to the hydraulic pumping action that can occur with a foam core. Second, it has better shear and compression strength, when compared with most foam core materials. Third, it has higher-thermal deformation resistance than most foam core materials. If the hull is an epoxy resin laminate with a foam core, it can start to deform under the heat generated by long exposure to sunlight if painted a dark color.
An epoxy-filled repair, even if it has a couple of bubbles trapped in it, will be much stronger than balsa could ever be. Properly done, it will exceed the life of the rest of the boat, even if the tang develops a leak in the future.
An epoxy-filled repair is usually far heavier and not as stiff as a properly designed cored-laminate of the same thickness. While thickened epoxy has great compressive and tensile strength, it doesn't have as much torsional strength as a properly cored laminate.
As for the damming technique I mentioned, I'm sorry if I wasn't too clear about how it's done. It's not done inside the core area, but rather on top of the deck. Might look kind of like a "volcano" or "funnel" of vinyl tape. Think of it as a "filler pipe" for the liquid. In fact, the initial epoxy repair can stand proud of the gelcoat, and then be sanded down flush before re-cutting the hole for the tang. End result is a solid plastic laminate, molecularly bonded to the internal balsa, and physically bonded to both the inner and outer skins. Very strong, and exceptionally waterproof. The nice thing about extending the form a little above the deck is that it provides a reservoir for the liquid epoxy. This allows gravity a little more room to float any bubbles up, and it also provides a reservoir so you don't have to refill the epoxy liquid as often during the time when it is soaking into the existing core.
I've prepared a lot of deck penetrations -- both new openings, and rot repairs -- with this technique, and they are all still in service. Some are decades old. I find this approach to be economical, easy, fast, effective, cosmetically invisible, and considerably better than the original construction.
While you may have done a lot of repairs using this technique, and they may still be in service, it doesn't necessarily mean that this was the right way to repair them. How many of those repairs involved an area as extensive as what the OP is discussing... In many cases, what is easy and fast, is not always the proper or right way to do something, and often far less effective than doing it the RIGHT way.
If the finished surface isn't going to be covered, you can even add pigment to the epoxy, though it does tend to darken with UV exposure. The technique has a lot going for it.
I wouldn't repair an entire rotted-core deck this way. The cost would be too high, evacuating bubbles would be difficult at best, and it would be a lot heavier than the original. In that case, I would use foam coring, bedded in thickened epoxy. But for fixing rotted deck penetrations, I know of nothing that beats this method.
If this had been caught earlier, I might agree with you that filling it with thickened epoxy would be a proper solution. However, this has gone far past a mere sealing of a rotted deck penetration at this point in time. Recoring the laminate there is the only proper solution IMHO.