A recent physorg article is titled "The measurements of the expansion of the universe don't add up". The article says
The current analysis of the variable brightness of cepheids with space telescopes such as the Hubble, along with other direct observations of objects in our cosmic environment and more distant supernovae, indicate that the H0 value is approximately 73.9 kilometres per second per megaparsec (an astronomical unit equivalent to about 3.26 million light years).
However, measurements based on the early Universe provide an average H0 value of 67.4 km/s/Mpc. These other records, obtained with data from the European Space Agency's Planck Satellite and other instruments, are obtained indirectly on the basis of the success of the standard cosmological model (Lambda-CDM model) ...
I thought I could learn more about these different methods from wikipedia, but the age of the universe article only talks about the Planck data and WMAP, and an age ~ 13.7B years. The wiki talk page does have a section that refers to the Hubble's Law page, which does talk about different methods and history, but also says
More recent measurements from the Planck mission published in 2018 indicate a lower value of 67.66±0.42% although, even more recently, in March 2019, a higher value of 74.03±1.42% has been determined using an improved procedure involving the Hubble Space Telescope. The two measurements disagree at the 4.4σ level, beyond a plausible level of chance. The resolution to this disagreement is an ongoing area of research.
So why is the Planck/WMAP estimate of the age of the universe preferred over similar alternatives, such as estimates based on Hubble observations?