I'mma disagree with y'all on the whole business of him should've had stop the stream or just replayed it another day or whatever.
There's a clear cut in the video listed and it's impossible to know how much time was between him saying he had to end the stream and when he came back after stopping his recording to finish it. He could've been gone anywhere from 30 seconds to 20 minutes to "console" her or whatever you're expected to do in that situation, and, honestly, what do you really expect? It's a dog he met maybe twice and it was 12 years old, so it's not like it's unexpected or tragic or anything.
Hell, most people can't even grieve properly, if necessary, immediately after someone close to them dies that's an actual human -- usually it's the day after when the shock of it all's worn off. And, yeah, I know that some people treat dogs like family members, or even get more worked up over them than if a friend or relative died, but it's not that huge of a deal. Better this than some overblown reaction like having to announce a miscarriage or that you have terminal brain cancer or some shit.