If the probability works out to 1 intelligent civilization per galaxy on average, then we wouldn't expect any other civilizations in our neck of the woods to discover (i.e. we could be the first or at least the only currently active one in our galaxy).

If the probability works out to a few per galaxy, we don't actually know that it's possible for a civilization to practically leave their own solar system. Or it could be that in the amount of time since the Big Bang, we'd be the only ones that have had time to develop in our galaxy since the Big Bang. Or we could be the only ones to have developed in our neck of the woods for distances that aren't horribly relativistic (e.g. our furthest telescopes are looking back billions of years so who knows the current state of those places).

As for the assumption that the galaxy would be teeming with life, it's not even clear that we can ever get to Alpha Centauri in a reasonable time frame. 80 years would require an average 5% the speed of light to get there (meaning much faster peak to do accelerate halfway / decelerate halfway). That's a lot of fuel so it's a huge stretch to claim we have that technology today. A 1% speed of life average speed would be a 400 year one-way trip. And even if we do build technology capable of this and go there, does that mean that Alpha Centauri then has the resources needed to bootstrap & proceed to the next star system or does it take a long time to build up a manufacturing base in the local star system to continue on the journey (i.e. you may be able to get to 1% the speed of light to travel between systems but traveling between planets within a system might still take a long time to collect resources & build out another space port)? In other words, how quickly do we go from star system to star system colonizing? There's no reason to believe that any intelligent civilization would be immune from economic pressures similar to humanity in terms of not taking on centennial or millennial interstellar projects. And as we can tell from humans on earth, there are economic, resource, technological, & timing reasons why we don't just start firing probes out into other star systems to try to do exploration.

We also don't know how long the universe needed to cool & coalesce for before life had the appropriate conditions to evolve; it's not like life existed from the first moment Earth formed. It likely needed to exist for billions of years after the solar system coalesced & then for billions of years it was still getting pummeled by asteroids & comets within the orbital plane until it got cleared out. After all, Mars seems to have been similar in many ways but ended up dead so it's clearly a knife's edge (e.g. Jupiter protecting us from a bunch of incoming asteroids and comets).

Finally, we have no idea whether intelligent civilizations have enough resources and survive long enough to leave their star system & to remain stable continuing to explore well beyond or collapse.

And all of that isn't even touching that our ability to look for these technosignatures is extremely limited because we don't know what we're looking for, we don't know where to look for it, & we are blindly guessing at the signatures we might find.

My point is - it's not a paradox unless you make some pretty wild assumptions that have no basis for them in the first place - we have no priors to inform probabilities to determine how many civilizations to look for. Nor do we even have search criteria for where to look. And we know that some pretty crazy technology that we have no way to detect is possible. As for visitation, I think people are way too optimistic about the ease with which such problems could be solved. & even if it does, we're at the outermost arm of the galaxy. Why would we be particularly interesting for aliens to visit? e.g. what star system would we pick to visit? Probably one with more resources that we've run out of rather than one with life if we've already convinced ourselves life is abundant in the universe & uninteresting to find.