Would you also argue that Junk Dealer and Upgrade should cost 2$ such that everybody could open with them?
This implicit comparison with Chapel is dubious for a simple reason: Chapel is a terminal whereas this is a cantrip. This is also why you rarely want more than one Chapel whereas you frequently want two cantrip trashers.
So besides being totally mispriced Nurse could imply that in a lot of Kingdoms the best opening is two Nurses.
I think there are reasons in favour of comparing Nurse to Chapel, and reasons against it. Obviously, allowing each player access to Chapel because it's good is something you could apply to Nurse. But i think the reason you listed weights stronger: You will open with two Nurses. In fact, i can imagine loading up on as many Nurses as i can get, simply because they don't harm your deck. Unlike Chapel, where multiples are actively bad usually, Nurses become better in multiples. Note this would still apply if Nurse was mandatory, as getting 2-3 Nurses would allow you to clear out your deck faster than one, and still can trash each other after that.
At the danger of sounding like a bad advertiser, take a look at my Homunculus ("Asper's Cards" thread, no link as i'm on mobile, sorry), which started very similar to Nurse. I gradually weakened it until it took its final form, where it is relatively easy to aquire, but delayed. Note that several other Alchemy cards, like Scrying Pool or Apothecary, and Travellers follow a similar pattern. This is to say, Nurse doesn't need to cost $5, but it certainly needs to be slowed down or made unattractive in multiples.