I don't see why the Enterprise couldn't be a Starship-Class, Constitiution-Class, and Heavy Cruiser-Class vessel.
This is a simple enough a rationalization.
It even gets amusing support from the other "TOS era" dedication plaque we (fail to) see: that of Kasidy Yates' transport Xhosa
. The vessel features a plaque of the same aesthetics and format, only with ANTARES CLASS in place of STARSHIP CLASS. And as we know, "Antares class" has through various mishaps become a more or less generic name for a variety of ship designs, and might thus have to be interpreted as a type description. A bit similar to "Panamax class" for a broad range of today's seagoing transports, perhaps?
Now all we really need is a glimpse at the dedication plaque of a TOS era Saladin
class ship, saying DESTROYER CLASS or perhaps STAR DESTROYER CLASS.
the term has been used far too widely since then for far too many types of ship.
But only long after it was used in the dedication plaque of NCC-1701. Back in those days, it got no competition from humbler ship types. So, terminology mutation.