Does he just happen to overhear the nurses talking about this in the hospital?
Yeah, this was a bit overkill, it was pretty unnecessary to overhear them for the MC to suspect there is something going on, the signs on the door would have been enough.
Both of them would have been fine on their own, but both are a bit unnecessary.
That said, I wouldn't consider the fact he happened to overhear the conversation to be bad writing.
Considering he got lucky enough to get reincarnated so it's not a huge suspension of disbelief he could get lucky again once in a while right ?.
Its just a plot hook for the MC to start investigating and it's not that strange for a nurse to question her actions so it's not out of place or anything like that.
The only sin committed here IMO is the fact that the author considered it necessary to have both plot hooks that basically fills the same function for the story and wastes space on the pages...
And the names being listed on the rooms despite them being very clearly proven to be empty beds? Like, what kind of mediocre hospital fraud is this?
Well, if you only enter the room on rare occasions, the visitors wouldn't question the fact that a patient is able to leave the bed and walk around.
Sure the person who has to stay in the room to get better might, but they also don´t see the name on the door as long as they are bedridden.
Once they are walking around they might also be considered well enough to leave soon and it's a minor thing most would overlook both as a visitor and as the patient.
The MC also knows that the owner of the place is a scumbag and is therefore also way more suspicious than the average person like his friend who doesn't notice anything is wrong.
And why the hell would they have these fake records in the room of the patients when the patients don't exist?
This is indeed a pretty unnecessary risk. even if you make the argument that what he found was a falsified doctor note for a checkup, its kind of unnecessary to write it on location.
The only real reason I can think of ( other then they being stupid for no reason ) is that the papers she needs to fill are clumped up together with the other patients of the same room for the sake of organizing the documentation and she was lazy.
This scene´s function in the story is to give the MC actually proof that something is wrong and not just a random well-placed hunch like last time at school.
There are other less stupid ways to do this, like have him break into the storage and find the fake documents there, but it works just as well even if I do agree its a bit hand fisted...
But on the other hand, even in real life people are usually a lot less smart and careful then they think they are.
Risk management never was humanity´s strong suit.