So Black Holes Actually Merge! In 1/5th of a Second - How?

This presumably stems from the fact that in the coordinate system of an external observer nothing can ever cross the event horizon of a black hole.

This is perfectly true, but if you were watching an object fall onto a stellar mass black hole it would red shift to invisibility in a few microseconds and it would look to you just like it crossed the horizon. More precisely, no matter how sensitive your measuring equipment there would be a time after which you could no longer detect that the object had not crossed the horizon, and for any physically reasonable equipment this time is extremely short.

The same principle applies to the merging black holes. We have two objects that can't actually be real black holes because in any finite universe we know real black holes cannot exist. However they are experimentally indistinguishable from real black holes. As these two objects approach each other the spacetime geometry changes and approaches that of a single rotating black hole - the Kerr metric. We know the geometry can never actually become Kerr because that would take an infinite time. However the geometry approaches the Kerr geometry so quickly that after a fifth of a second it is experimentally indistinguishable from the Kerr geometry.

Whether the black holes have merged or not depends on exactly what you mean by merged. They are certainly no longer two separate objects, and that happens in a short time and is observable. In this sense it seems reasonable to me to describe tham as having merged. If you insist the merger is complete only when the transition to the Kerr geometry is complete then this will take an infinite time so they will never merge.

tl;dr - in any sensible meaning of the term merge the two black holes do indeed merge in a finite, and very short, time.