I believe you, but I'm curious how that works. When you go to a random doctor, do they have to request your records from all your other doctors? Similar to here in the USA when you have a PPO?
One, in some of the countries I know (with universal healthcare and no centralised records) you don't go to a random doctor. You have a declared family doctor and you have to go to them unless they are unavailable, in which case the other doctor you go to has to declare that you couldn't go to your doctor. It's a small hurdle to prevent doctor shopping, but it means people are more likely to always see the same doctor. Specialists are given the relevant information by the family doctor when referring a patient to a specialist, and in most other cases records are not really needed, or the ER will contact whoever to get the information they think they need. It might sound hazardous but in practice it works fine.
Second, some places have centrally-stored records but the access is controlled by the patient. Every access to the record is disclosed to the patient and he has the possibility to revoke access to anyone at any time. That generally goes together with laws that fundamentally oppose any automated access or sharing of these records to third parties.
And third, I don't understand what any of this has to do with who whether healthcare access is universal or not? Universal healthcare without centralised records exists (in France, unless it has changed in recent years, but it at least existed for 60 years or so) and centralised records without universal healthcare could exist (maybe privately managed by insurance companies, since the absence of universal healthcare would indicate a pretty disengaged state).
Universal healthcare is about who is paying, not necessarily about who is running the service.
This was somewhat annoying since unlike the UK system, the Australian system is essentially private GPs getting paid for your individual appointments by the government (so called bulk billing), so there's no guarantee that you can go to the same doctor every time.
Basically, government funded and regulated doesn't mean government run.
There is no standardized EHR system here, despite provincial governments (which are who runs the systems) wasting millions over the last two decades trying to make that happen.
This was the last decades way of doing things. The current decade is to stay within the desired charting system. That way you can one-click share data between doctors. Typically you would search for doctors that utilize the same charting platform. EPIC is probably the largest one in US today