If one defines "radical" as "extreme," then the radical republicans were indeed radical. They were determined to remake the South as a model of the North with no regard to the history and culture of the South, which they considered anachronistic and inferior. The position of Abraham Lincoln and other Republicans before and during the war had been that the southern states had no authority to leave the union, but had rather simply been in rebellion. The Radical Republicans all but abandoned this position to make sure they were in complete control of reconstruction. Thaddeus Stevens of Pennsylvania argued that the former Confederate states had reverted to "conquered provinces."
It is obvious from all this that the first duty of Congress is to pass a law declaring the condition of these outside or defunct States, and providing proper civil governments for them. Since the conquest they have been governed by martial law. Military rule is necessarily despotic, and ought not to exist longer than is absolutely necessary. As there are no symptoms that the people of these provinces will be prepared to participate in constitutional government for some years, I know of no arrangement so proper for them as territorial governments. There they can learn the principles of freedom and eat the fruit of foul rebellion. Under such governments, while electing members to the territorial Legislatures, they will necessarily mingle with those to whom Congress shall extend the right of suffrage. In Territories Congress fixes the qualifications of electors; and I know of no better place nor better occasion for the conquered rebels and the conqueror to practice justice to all men, and accustom themselves to make and obey equal laws. . .
Charles Sumner of Massachusetts argued that the Southern states had committed political suicide:
Any vote of secession or other act by which any State may undertake to put an end to the supremacy of the Constitution within its territory is inoperative and void against the Constitution, and when sustained by force it becomes a practical ABDICATION by the State of all rights under the Constitution, while the treason it involves still further works an instant FORFEITURE of all those functions and powers essential to the continued existence of the State as a body politic, so that from that time forward the territory falls under the exclusive jurisdiction of Congress as other territory, and the State, being according to the language of the law felo de se, ceases to exist.
The Radical Republicans were then radical in every sense of the word, unwilling to concede or compromise in any manner with the former Confederate States.