Explain how some progressive mayors and governors became even more politically entrenched than the machine politicians that they replaced in office.

Cermak defeated Thomson in mayoral elections in 1931 mayoral elections and died during the assassination attempts over the elected president Franklin D Roosevelt. Patrick A. Nash led the appointment of Edward J.Kelly as mayor. The Kelly- Nash authoritarianism just followed Cermak's machine rule, however, Kelly- Nash had political appointments and many favors to specific ethnic groups.
He kept the city solvent through the liberal use of federal funds at a time when the Great Depression provided the most serious threat to the financial prosperity of municipal governments; he got additional financial support from organized crime. Concerns after Kelly’s reelection were about the number of scandals in municipal administration and seriously in the public school system surfaced alongside a increasing public outcry against the his organized crime in the city. The party requested Kelly not to seek reelection in 1947 and replaced him with Martin H. Kennelly.
Kennelly's presence as a mayor office for two terms was due to his democratic machine rule, but then replaced him with Richard J. Daley, in 1955. During Daley's prolonged tenure in city hall he was reelected five times before to his sudden death in 1976. At a time when virtually no urban political machines survived, Daley led the Cook County from one election triumph to another. When workers died or retired, he filled their positions for the moment pending civil service exams that were never given.
By the 1980s, the mighty Democratic patronage army reduced greatly, but during the Daley years civil servants who worked hard for the party at election time and the area captains who produced healthy victory margins at the polls kept their patronage jobs and received some rewards. Chicago was prominent during Daley’s term as a mayor. During Daley’s term he helped O’Hare international Airport construction of the Illinois university branch campus, expansion of the city's road network.
The campaign promises to the contrary, Byrne did not put into consideration black political demands. Then the election of Harold Washington as the city's first black mayor in the 1983 and his subsequent reelection four years later explicitly ended Democratic machine rule in Chicago. The election to the mayoralty position of Richard M. Daley, the eldest son of the deceased boss, did not indicate a resurrection of the machine in a new appearance.
The younger Daley readily acknowledged, totally different demographics and the attendant alterations in the political calculus clearly made the machine politics for which Chicago became famous leftover by the end of the twentieth century.
Conclusion
People rejected the mayors that showed authoritarianism in their leadership or those called the machine mayors whereas those that addressed the needs of the people were voted for renewed terms in office.

Biles, Roger. (1995). Richard J. Daley: Politics, Race, and the Governing of Chicago. Erie, Steven P (1988). Rainbow’s End: Irish-Americans and the Dilemmas of Urban Machine Politics, 1840–1985.