Why is the Planck/WMAP estimate of the age of the universe preferred? 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.[60] The two measurements disagree at the 4.4σ level, beyond a plausible level of chance.[61] The resolution to this disagreement is an ongoing area of research.[62]

So why is the Planck/WMAP estimate of the age of the universe preferred over similar alternatives, such as estimates based on Hubble observations?
 A: *

*Hubble observations can only produce a local measurement of Hubble
constant $H_0$. The cosmic ladders such as cepheids and supernovae
are used to arrive at the Hubble parameter $H(t)$ at low redshift. In other
words, Hubble observations only tell you the cosmic behavior of the recent
epoch. It's a snapshot of the universe in its adulthood, with beards and what not. 

*Planck Satellite observations determine cosmic parameters (6 of them) based the remnant signals emanating from the universe in its embryonic stage, such as Cosmic Microwave Background (CMB). It's a point-in-time picture of the universe when it's still a cute baby with chubby face. 


Two takeaways:


*

*Since Planck Satellite observations are interpreted in the context of $\Lambda CMD$ to reproduce the whole evolution history of the universe, one can impute the age of the universe as well as the current value of the Hubble parameter $H_0$. Nevertheless, one should bear in mind that these Planck results are model ($\Lambda CMD$) dependent. Given the tightened error margin in cosmic ladder and CMB measurements, the $H_0$ tension between Hubble and Planck is most likely due to some defect in the standard cosmology model $\Lambda CMD$. Therefore the Planck-based estimation (in conjunction with $\Lambda CMD$) of the age of the universe is questionable.

*Hubble observations CAN be used to directly calculate the Hubble constant $H_0$ without resorting to any cosmological model such as the concordance model $\Lambda CMD$. However, without enlisting the aid from $\Lambda CMD$ (and the questionable $\Lambda CMD$ parameters sourced from Planck), Hubble observations alone are INCAPABLE of inferring the age of the Universe.


Is there any direct way of observing the age of the universe? Well, the estimation of the age of the old globular clusters might put a lower bound on the age of the universe, since the universe must be older than these globular clusters. As of yet, the observations of the age of the oldest globular clusters are inconclusive, with some of them MIGHT exceed the Planck-based estimation of age of the universe.
And this is where things stand with the wonderfully "accurate" standard cosmology. 
A: You have partly answered your own question. The Planck measurements are more precise. Moreover, it is generally assumed, perhaps wrongly, that the measurements based on the cosmic microwave background are subject to fewer systematic uncertainties than other methods.
You also need to think about how the age is derived. For that you need more than just the value of $H_0$. In particular you need estimates of the energy densities of matter and dark energy. These can come from the cosmic microwave background data alone and give the value of 13.8 billion years.
If $H_0$ from Planck and from other methods do not agree, then there is a problem with the measurements, cosmology, or both. But you can't really just adopt a different value of $H_0$ and then use the other cosmological parameters that are derived from the CMB to derive a new age (though people have).
