If objects approaching the event horizon of a black hole appear to slow down to outside observers, how was LIGO able to "see" black holes collide? Wouldn't their collision appear to stop as their event horizons met?
There are a lot of Q/A's on the BH mergers, maybe even one or two on your same question. It still always brings up incredibility in the eyes of anyone thinking about it.
The simple answer is that you can think of BHs as having 'almost-horizons'. Ie, as they were formed astrophysically, the got to form their horizons maybe within a few thousands or millions (or more or less, one can calculate it as function of time) of Planck lengths, and to any outside observer it would have looked like a BH. When two of them merge its the same effect, the two horizons get so close to each other that the effect is that they look like they merged, and in the process form one 'almost-horizon', shedding all the hair to look like a No Hair BH.
Some people say they have apparent horizons. Apparent horizons have been studied rigorously, see their basics and references in wiki at https://en.m.wikipedia.org/wiki/Apparent_horizon. Most of that work, and related similar terms, are to study dynamic horizons, how they form and react to disturbances. Hawking said not too long ago that there are no BHs, most people think for this, and maybe quantum gravity reasons. Nevertheless, practically, we can treat them as BHs, they are indistinguishable from those.
This is closely related to the question Can you have a giraffe shaped black hole? that I answered yesterday.
You are quite correct that objects take an infinite time to reach a black hole, let alone pass through it. However they take only milliseconds to get so close to the event horizon that they appear to have merged with it.
In the case of a black hole merger the two initial objects weren't really black holes because it takes an infinite time to form the event horizon. And the final merged object isn't really a black hole either, for the same reason. However the approach to the horizon is so fast that within a few milliseconds the merged object is indistinguishable from a true black hole.