Personally, I think it's highly likely that if dinosaurs were re-introduced into the world, that they would be ill adapted to current conditions and would mostly go extinct again. I would note that there has been plenty of time for lizards and/or birds to evolve to larger size since the time of the dinosaurs, but they haven't. Large land-based birds have mostly survived only in small niches like on Australia.
Animals rarely go extinct because they were outcompeted by a new animal out of nowhere. Once an animal finds an ecological niche, it tends to stay there. What bumps animals out is almost never some upstart with better adaptations, but an environmental change. Evolution, after all, is not the abstract process of generalized improvement that it's often made out to be. It's about adapting to a specific set of environmental conditions. When there's a significant change, like rising temperatures, or the emergence of grass or angiosperms, there can be an almost complete overturn of the faunal assemblage. The most extreme examples, of course, are extinction events, which essentially clear the board and allow new forms to colonize most of the now-vacant niches. That's what happened when the dinosaurs died. Most of the time it's a smaller, more gradual process. A river drying up, a toxic new species of algae moving in, or jellyfish eating all the baby fish; that kind of thing.
This ties in with punctuated equilibrium, the idea that, for most of the fossil record, species don't change much. But then there are sudden bursts, where rapid change occurs. That happens when the environment changes, creating new niches or destroying old ones.
So the first question to ask isn't whether raptors can beat lions in the hunt for gazelle, but whether dinosaurs have the adaptations necessary to survive in today's environments. That's a complicated question. But it's not about whether mammals are more advanced in some generalized way. First of all, that's an inaccurate way of looking at the evolutionary process. Barnacles and sharks still thrive, even though they're considered "primitive". They succeed because they have a complex suite of adaptations that are well suited to the environment in which they live. And the environment has changed a lot in the last 66 million years. Newly reborn dinosaurs might die out because little mammals eat all their eggs, or because they can't handle C4 grasses, or because PH levels are wrong, or a million other subtle things.
But let's assume at least some species can overcome that hurdle. Will raptors beat lions? Still hard to say. One of the situations where animals do directly compete with each other for the same niche is during a faunal exchange. There have been many in the past, as previously isolated land masses came into contact with each other, or other natural barriers eroded. For instance, the period during the Miocene when Afroarabia started to connect with the European archipelago and Asia. Or for a recent and more famous example, the Great American Interchange, when the Ithmus of Panama rose and connected North and South America. What happened? The fauna of the two continents merged, but... it wasn't a balanced mix. Basically, South America, with the giant birds and marsupials, lost, and lost hard. Exactly one marsupial descended from that exchange still exists in North America, the 'possum.
Why? Well, that's also hard to say. It's a huge array or species with a huge array of adaptations adapting to a huge array of environments, and the smallest thing might be what makes it thrive or die. But one thing is consistent: The winner in exchanges is usually the fauna from the largest landmass. In this case, North American fauna had crossed the Bering Straight from Afroeurasia. Their ancestors were the winners on that highly competitive supercontinent. In contrast, the South American animals were mostly emigrees from Australia, which had much less room for fierce competition.
Which brings us to birds. You're talking about birds as if they've been tiny since the age of dinosaurs, but that's wrong. One of the most recurrent patterns throughout the Cenozoic is avaian gigantism. It happened again and again, and seems to be a consequence of bird physiology. Birds are highly energetic, and flight is a harsh mistress. So they're ruthlessly adapted for low weight, and when they no longer have an evolutionary need to fly, the first thing that happens is they shed all those adaptations, at an extremely rapid rate. This means all those constraints that keep them small are thrown away, and they often become quite large. Combined with their ability to fly around the world, across environmental boundaries like the Wallace line, what happens is they're able to colonize the most remote habits, and then quickly adapt and become large predators.
But they're just adapting to an island, with an island's tiny land area, so when the island is colonized by competitors from a nearby continent, they tend to quickly be outcompeted and go extinct. This isn't anything essential to the nature of birds vis-a-vis mammals. It's more that birds are opportunistic colonizers, able to reach and quickly adapt to new environments, but they lack the time and evolutionary pressures of a large land mass, so when competitors from those larger realms reach their shores, they tend to get wiped out.
Pat and S'mon -- you each claim I'm wrong because there was only one shift downwards is size ...
Nobody said that. We both provided a counterexample, neither of us claimed it was the only instance. You were arguing that there is a consistent trend for animals to get smaller over time. That's false. There is no such trend.