The only thing wrong with a hospital ship is that by its very mission it's supposed to stay out of danger and avoid
getting involved in situations where a show of force is necessary. Essentially, you'll have E.R.
or Grey's Anatomy
in space, which might be interesting for some
folks, but probably boring for the majority of Trek fans who'd rather have something more action-oriented rather than a medical drama with some aliens tossed in.
The alternative is to have the hospital ship be frequently put in dangerous situations and saving [INSERT NAME HERE] from an interstellar disaster--but in such a case, the difference between it and a standard hero ship is academic.
Both VOY and ENT took the exploring space too literally by having their hero ships be generally alone out there by themselves, which can often lead to a lot of repetitive alien-of-the-week stories. TOS and TNG simply had exploring be an excuse for why their ships were out there, but they really told a wide variety of stories in which the actual exploration was investigating something amiss in the Federation's own backyard.