You generally don't. If it pass a read/write test, that is probably about as good as you can get.
Have good backups, and redundancy, and don't worry about it.
We are digital librarians. Among us are represented the various reasons to keep data -- legal requirements, competitive requirements, uncertainty of permanence of cloud services, distaste for transmitting your data externally (e.g. government or corporate espionage), cultural and familial archivists, internet collapse preppers, and people who do it themselves so they're sure it's done right. Everyone has their reasons for curating the data they have decided to keep (either forever or For A Damn Long Time (tm) ). Along the way we have sought out like-minded individuals to exchange strategies, war stories, and cautionary tales of failures.
You generally don't. If it pass a read/write test, that is probably about as good as you can get.
Have good backups, and redundancy, and don't worry about it.
With a single disk probably impossible.
But if you have the same HDD already and know it works reliable for some time then you could compare (run them in an open USB case or something during that). Also a good idea to gently touch them, so you can 'feel' the respective vibration patterns (good idea to wear an anti-static arm band thingy).
If the new disk —after some "warming up time" (e.g. after it has run for about a week. Use that time to do the zeroing out and SMART long test!)— is much louder, makes unusual clunky noises, scratching noises, high frequent peeps, or other strange sounds much worse than your old disk then may be worth exchanging it as precaution.
Basic mechanical tests are done during the SMART tests (e.g. if you run a short test you'll hear).
Basically, I don't think there is a way to check that. Usually, with the new drive you get a warranty, so even if it fails in some time you get the replacement.
Do not forget about backups since even healthy drives can fail at any moment for any reason, so you have to be prepared for that.
Hard drives only write and read data, and everything is tested when you do such operations. It's not like the engine is working but not the radio or I didn't test the WiFi and the antena is busted or something.