First in the Milky Way? I think that's a stretch. First in this region? Possibly.
I mostly share your view, but I'll add that we actually know one of the relevant variables – the number of stars in the galaxy and/or universe. We don't actually know the probability of life starting, or the probability of any life evolving into species that form an intelligent civilisation, or how probable is it for the process to fit within the lifespan of a star with a habitable zone. We reason that it must be somewhat probable, because we have kind of sketched the process, but we don't really know. If the overall probability comes out very very very low, we could be the only life in not just the galaxy, but the whole universe.
It's crazy, but our existence in itself is not evidence of anything, since we would experience nearly the same whether civilisations like ours were found once in 100 stars, once in a galaxy, or once in 100 universes. The only difference would come with better understanding and knowledge of evolution and abiogenesis from studying Earth life, to the point where we can tell how likely we are (and therefore everybody else is).
It will be kind of creepy if we learn this before we get direct evidence. We might be at a point where we have scientifically proven that there is other intelligent life in the galaxy, but we are still nowhere in detecting it, every thread like this one is still a false alarm and we have little hope of finding anything.
I've been wishing for us to find find life – even extinct past life – elsewhere in the Solar system (Mars, Enceladus, Europa), as that will solve the first part of the problem the easy way. Well, unless it's related to Earth's life – then it wouldn't be as helpful.
(Other than that, I'm somewhere between “only life in the universe” and “there are clay pots on Mars”, with personal preference of nearest life about 10 light-years away, and nearest civilisation about 500 light years away.)
ETA: There's one argument for being alone in the galaxy though – the possibility for a civilisation to settle the whole thing by launching colonies exponentially.