>>The group containing frogs and birds has one common ancestor and all of the descendents except for three groups of species (i.e., urodeles, gymnophiona, and amniota). Isn't that paraphyletic in the same way reptiles are paraphyletic? Reptiles have one common ancestor and a few groups of descendant species removed (e.g., synapsida and aves).>>

Tetrapods include more than the groups you cite. There are basal amphibians that are included among tetrapods besides the urodeles, and the caecilians. You are incorrect that your group (birds plus frogs) excludes the amniota because birds are included in amniota. If you were to exclude all amniotes from your taxon, that leaves frogs only, which is of course a monophyletic group. Frogs do not even form a paraphyletic group. It is a holophyletic group.

Reptilia is a monophyletic group (paraphyletic according to the fastidious cladists) because it shares a nearest common ancestor. Since all life on earth share a single common ancestor, any 2 or more species on earth can therefore form a monophyletic group. Hence there would be no such thing as a polyphyletic group using such a broad definition of monophyletic.

Biologists have agreed that a monophyletic group should be defined as a group in which all members must share a nearest common ancestor. Reptilia is a paraphyletic group (according to the fastidious cladists) because it is a stem group within Amniota. The lineages of all of the members of Reptilia can be traced directly to the ancestral stem species of Amniota. Reptiles is paraphyletic (instead of holophyletic) because two crown groups of amniotes have been removed: namely the birds and the mammals. Since the birds have been removed from Amniota, the many lineages of the Aves cannot be traced directly to the common ancestor of the amniotes. The nearest common ancestor of Aves is now the first archosaurian reptile with feathers (Longisquama being the oldest known reptile with feathers would be a good candidate as bird ancestor, not withstanding many cladists' denial that Longisquama has feathers because they dogmatically believe that birds are descended from a dinosaur and Longisquama is not a dinosaur). The nearest common ancestor of the mammals is a therapsid reptile that evolved fur from whiskers. Lumping two crown groups like birds and mammals would create a polyphyletic group because these two groups have different nearest common ancestors. Lumping frogs and birds into the same taxon is similar to lumping mammals and birds because they too have different nearest common ancestors.

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