If I understand correctly, I think his claim is based on his understanding about the probability of single cellular organisms evolving into multi-cellular organisms. But I don't have access to the data he may have used to calculate that.
However, that particular evolutionary step would seem to be critical in the evolution of technological intelligence (at least if we keep our view of "life" to the narrow band of water based organics which we are familiar with), so if for some reason the probabilities of that happening are small enough, then the stability period for planets or moons may be too short to allow for it in a single Galaxy.
I would also point out that he referred to the meteor event which wiped out the dinosaurs, so i think he was not solely referring to the step from single-cell life forms to multi-cell life forms. Either he did not express himself well, or, as i think is more likely, the journalists were left far behind right at the starting gate, and most of what they're reporting is considerably less than coherent.
Your point about the stability of planets or their satellites is well-taken. I think it is far less likely that stable planets/satellites would form and endure in the galactic center. We see gamma ray bursts all over the sky. Where stars are more densely clustered, the probability of potentially disastrous events is far greater. You could have a planet with multi-cellular life or even sentient, self-aware life forms reaching or already possessing technology which gets hit by the shock wave of a star going nova, which strips off the atmosphere. They'd have just about enough warning to bend over and kiss their respective asses goodbye. The gravitational tidal forces in the center of the galaxy could be such that planets do not easily form, or don't last long if they do.
Of course, we are in the galactic boondocks where such dangers are of a much lower probability. It could be that it is only on the periphery of galaxies that the rise of multi-cellular life and in particular, sentient, self-aware species is of a high order of probability. The so-called cinderella zone is, in my opinion, much over-stated. You probably need liquid water for higher complexity organisms, and for technology, you probably need sufficient oxygen for fire. So you also need dry land. (There are likely not many occasions for starting fires on water worlds. If you've got an ocean world, you could have a sentient, self-aware species such as dolphins or whales--if that is a realistic description of those creatures on our planet--and you're not going to be coming up with much in the way of durable technology.) You probably need the equivalent of fingers and thumbs.
One of the issues which that silly Fermi paradox never considers is social organization. Even if you have a sentient, self-aware species, a crucial question is going to be how well the work together, and whether or not they would ruvive the rise of technology.
I would be very much interested to know the deatils of his reasoning.