There are strong arguments for central bank independence from our best understanding of macroeconomics. Not only central bank independence insulates policy from the temptation to exploit political business cycles but it also protects against the temptation that government finances its spending via monetization (e.g. see discussions in Friedman 1962 or Drazen 2002). Both of the issues above can have disastrous consequences for an economy, and monetary policy is extremely important and can cause great harm if mismanaged (for example Great Recession was to great extent caused by bad monetary policy).
Second there is empirical evidence that independent central banks perform much better at their job of keeping price stability. For example, the seminal work of Cukierman (2010) and Alesina and Summers (1993) show that independent central banks perform much better than the ones that are not independent (although there is some new work that tries to challenge this, e.g. see De Haan et al 2018, it is still consensus that independent central banks perform better).
Also, as summarized in the review of literature on central bank independence by Wachtel & Blejer (2020):
In summary, CBI [central bank independence] moved to the forefront due to four factors: (1) reactions to high inflation experiences; (2) interest in central bank legislation and constitutions and the establishment of many new or reconstituted central banks; (3) developments in macroeconomics, particularly rational expectations and time inconsistency; and (4) empirical evidence. All of these combined to build a convincing case that CBI is essential to constrain political influence and provide sound monetary policy. By the turn of the century, there was a strong consensus view among economists, central bankers, and governments in support of CBI. Central bankers found that CBI gave them the ability to ignore criticism and maintain policies consistent with long‐run objectives.
Furthermore, as further argued by Wachtel & Blejer (2020), contemporary central banks have roles that go beyond monetary policy such as conducting financial oversight or monetary research and these are also best done without political meddling (e.g. even many government run regulatory agencies (like the U.S. Securities and Exchange Commission) and research universities are independent precisely because it is recognized that political meddling in research or day to day regulation is detrimental in its own right).