It is awful, though the main problem with the system is in the financing of care, as far as I can tell. You would think that would be easier to fix than developing a rather more advanced healthcare system, because there are already very qualified professionals, equipment, etc.
In any case, there's no centralized database in the US as well. From the CDC website (where the "unfortunately" is in response to locating lost records):
> Unfortunately, there is no national organization that maintains vaccination records. The CDC does not have this information. The records that exist are the ones you or your parents were given when the vaccines were administered and the ones in the medical record of the doctor or clinic where the vaccines were given.
https://www.cdc.gov/vaccines/adults/vaccination-records.html
> People could look back in 100 years and identify the coronavirus epidemic as the moment when a new regime of surveillance took over, especially surveillance under the skin ... is maybe the most important development of the 21st century, this ability to hack human beings, to go under the skin, collect biometric data, analyze it, and understand people better than they understand themselves. This is maybe the most important event of the 21st century.
From Dec 2019, https://news.mit.edu/2019/storing-vaccine-history-skin-1218
> MIT researchers have now developed a novel way to record a patient’s vaccination history: storing the data in a pattern of dye, invisible to the naked eye, that is delivered under the skin at the same time as the vaccine ... By selectively loading microparticles into microneedles, the patches deliver a pattern in the skin that is invisible to the naked eye but can be scanned with a smartphone ... “It’s possible someday that this ‘invisible’ approach could create new possibilities for data storage, biosensing, and vaccine applications