I think the Prometheus acquitted itself rather well, taking only minimal damage while fighting a small fleet of both Romulans and Feddies (including one Defiant-class!), crewed only by two bumbling medical holograms
Sisko hints at the Defiant's secretive nature in "The Search":
"Offically, she's an escort. Unofficially, the Defiant's a warship. Nothing more, nothing less."
Shouldn't be a big surprise that Starfleet didn't necessarily want word out that they had built a ship to be a straight-up brawler, not to mention it had a cloaking device too.
If the Defiant is unofficial considered a warship, shouldn't the Galaxy class, Nebula, class, Akira class, Prometheus class, and Sovereign class starships be, too?
The Defiant was designed as a warship, and only a warship. Everything else you mentioned had other roles, ones which could lead them to having to defend themselves (this goes triple for the more exploring types); since Starfleet seems to value their ships and crews, they would want them to be able to defend themselves as well as possible. Even the designation of "escort" implies a defensive role, and not even necessarily a highly-armed one (defending freighters or VIP's from pirates, etc). But warship implies offensive power, and no other purpose.