The best resource I've found that describes the shortcomings of the previous definition of the kelvin is
The kelvin redefinition and its mise en pratique. B Fellmuth et al. Phil. Trans. R. Soc. A 374, 20150037 (2016).
Basically, the pre-2018 kelvin, defined via the triple point of water and propagated to the full range of temperatures via the ITS-90 scale, is the best we have for most temperature ranges, but it can be improved at very low and very high temperatures:
While the redefinition of the kelvin will have no impact on the status of the ITS-90 or PLTS-2000, there will be significant benefits, particularly for temperature measurements below approximately 20 K and above approximately 1300 K, where primary thermometers may offer a lower thermodynamic uncertainty than is currently available with the defined scales.
Here primary thermometry refers to the direct measurement of thermodynamic temperature, as defined in statistical mechanics (but see the paper's §3 for a more precise definition). This can be done via a variety of methods but, with current technology, those methods are only able to achieve better stability, precision and reproducibility at the < 20K and > 1300 K ranges mentioned above. In the middle interval, the change in definition does not affect practical thermometry:
In particular, the most precise temperature measurements in the core temperature range from approximately 25 to 1235 K will, at least initially, continue to be traceable to standard platinum resistance thermometers calibrated according to the ITS-90.
This leaves us, then, with two ranges where there are primary-thermometry methods that beat the ITS-90 precision:
At very low temperatures you can use Acoustic Gas Thermometry, where you have a dilute gas that you can treat as an ideal gas, which is an extremely well-characterized system. Here you measure the speed of sound, which depends on the temperature in well-understood ways.
At very high temperatures, radiometric thermometry uses Planck's spectral law to infer the temperature of a glowing object from the spectrum of the electromagnetic radiation that it emits. This is again a well-characterized system where the thermodynamic temperature enters only via $k_BT$ to the measurable observables.
Looking forward, there also seems to be an expectation among metrologists that the pace of improvement in precision and reproducibility in primary thermometry will continue to beat that of practical temperature scales, which means that $k_B$-based thermometry will eventually displace ITS-90 in other temperature ranges. This future-proofing is an important aspect of the change to a universal-constant-based definition.
For now, though, only those two extreme temperature ranges are affected.