UUIDv4 removes all three of those vectors. UUIDv7 still removes two of three. It doesn't leak record count or the rate at which you create them, only creation time. And you still can't guess adjacent keys. It's a pretty narrow information leakage for something you routinely reveal on purpose.
I can see it being bad for tracking IDs, but not order IDs, unless you are allowed to view any orders that do not belong to your account, which is just fundamentally bad security and using UUIDv4 or a random string would simply be obscuring security.
With UUIDv7 the creation time is always leaked without any sampling. A casual attacker could quite easily lookup the time and become motivated in probing and linking the account further
When sequential integer ID's are externalized, an attacker does not need creation times to perform predictive attacks. All they need to do is apply deltas to known identifiers.
Always thought that was elegant (the attach not using the time as the seed).
You wouldn't be publishing patient visits publically, the only folks that'd legitimatly see that record would be those which access to that visit, and they'd most likely need to know the time of said visit. This access should be controlled via AuthN, AuthZ and audited.
You'd also generally do a lot of time-based lookups on this data; what visits do I have today, this week, and so on. You might also want an additional DateTime field for timezones and offsets, but the v7 is probably better than v4 for this usecase.