How about a “yahbut” on this?
Yes, there needs to be a clade that includes the common ancestors of mammals, modern reptiles, and birds, and it’s referred to as Amniota – all those vertebrates that produce a fluid filled protective membrane containing liquid, within which the embryo can grow, usually accompanied by a yolk or other food source, and hence can be laid as eggs that do not have to develop into tadpoles in the water. (Mammals, of course, retain their “eggs” within the body and, in lieu of yolk, develop the placenta to nourish the developing embryo from food assimilated by the mother – but we retain the amnion, the “bag of waters” within which the baby develops.) Amniotes and living and extinct amphibians share a common ancestry and set of characteristics – four limbs, lungs for breathing air, etc., in the clade Tetrapoda. Another clade unites Tetrapods with lungfish and coelacanths and the extinct rhipidistians in having two sets of paired appendages containing bone and muscular flesh for use in locomotion, as well as several other adaptations.
The point here is that there are a shitload of kinds of bats, adapted for a wide range of aviational lifestyles: insectivores, fruit-eaters, pollinators, haemovores, etc. They have a common ancestor but have radiated into a number of roles.
Nobody has an issue with bats being mammals. But you can identify a bunch of characteristics that unite shrews, rabbits, oxen, bears, elephants, monkeys, and armadillos that do not describe bats – they’ve specialized for flight.
Likewise, dinosaurs radiated over a wide range of characteristics, including small carnivorous, insectivorous, herbivorous, and fructivorous roles as well as the ones better known because they were huge and intriguing. Like the chiropteran mammals, the aviform dinosaurs conquered the air and radiated widely.
But, unlike the present day, where mammals are widespread, the only surviving archosaurs are the crocodiles, distant cousinsof the dinosaurs, and the birds, the group of dinosaurs which adapted for flight-based lifestyles.
Imagine a mass extinction event that kills off almost all warm-blooded tetrapods. Lizards and snakes survive. And so do some bats, on remote islands protected from the mass extinction by distance and isolation. The didelphids (opossum family) live through this one the same way as they did through the K-T one, by conservatively hanging to an opportunistic omnivorous lifestyle.
Lizards radiate to fill most land-animal niches. Possum descendants take a few niches, but remember that they’re conservative in evolutionary terms. And bats radiate widely to fill the niches once occupied by birds. It’s easy to see the relationships between the great-hunter eagle-bat, the white-furred long-legged egret-bat, the conservative acrobatic swift-bat, the diving mallard-bat, etc. But it is very difficult to conceive that the great panoply of Chiropterans were ever closely related to those strange extinct creatures like the rhinoceros, the elephant, the Kodiak bear, etc. – they seem nothing alike.
That hypothetical describes very well our problem with dinosaurs and birds. Say “dinosaur” and people think Brontosaurus, Tyrannosaurus, Triceratops – not, as a rule, Coelurus or Psittacosaurus. But there are some pretty clear transitional forms between the smaller raptors (Coelurosaurs) and the early birds – Shenzhouraptor and Yandangornis are typical of these “feathered raptor” sorts of creatures.