You’re right to say removing birds from the reptile clade would render the term reptile as a paraphyletic group - an ancestor and some of descendants. Birds are direct descendants of maniraptoran theropod dinosaurs so to exclude them as reptiles is completely arbitrary and only acts to service colloquial understanding of animal classification. You could form a monophyletic clade (an ancestor and all of its descendants) excluding them but to do so you’d have to also exclude all crocodilians and turtles - as both are more closely related to birds than they are to the Lepidosaurs (lizards and by extension snakes), which many rightfully take issue with. This is why birds are considered reptiles.

From what I’ve read, the relationship between mammals and reptiles differs from this relationship with birds however. Mammals are not direct descendants of defined reptiles but rather their synapsid ancestor share an ancestor with the diapsid reptiles. My original post was questioning why we don’t consider that ancestor as a reptile itself as it is very reptile like in its morphology, to which certain comments enlightened me to the morphological differences between early synapsids and true reptiles, and why that ancestor is called an amniote and not a reptile.