Did Starfleet build warships?

Agreed. I do think that perhaps there are a handful of designs, like the dreadnoughts and the Defiant, which might be considered to be true dedicated warships. But many of the sources I've read which have included such vessels have described them as being a minority of vessels, given the increased resource costs and the fact that they tend to be less suitable for more traditional roles like exploration. I think the majority of designs we see are multipurpose, as is probably true of many alien ships as well.

The Excelsior was considered a battleship in the FASA RPG, and I think it could have easily been built as such initially and then perhaps downgraded to a heavy cruiser, as it's referred to in other sources. It was intended initially to be able to face threats from the Klingons and Romulans in the movie era.

 

I'd like to think Starfleet wouldn't be quite so quick to decommission large groups of designs that readily, even if it no longer produces them. If they're still using Excelsiors and Mirandas effectively nearly a century after those designs were considered reasonably new, it stands to reason that there would still be Constellations, etc. in service as well. It's very difficult to judge how long an average design life span is, but unless it has a glaring design problem, I see no problem with it lasting that long.