“The federal government never recognized secession as legal but made all the Southern states apply and be readmitted to the Union.”
That’s not true: They did not have to be readmitted, as their secession was declared to be illegal, in which case they had never left.
Texas fully rejoined the Union on March 30, 1870, when President Grant signed the act to readmit Texas to Congressional Representation. “The United States government has never recognized the right of states to secede, and considers the states to never have left the union during the American Civil War. Yet, the states were required to agree to Reconstruction before being permitted to send representatives to Congress again.”