5e seems to make room for both. I think. *crosses fingers*
Using your definitions I don't see why it doesn't, or really, how it
couldn't. In truth,
any edition can facilitate both "adventurous" and "heroic" campaigns - this mainly has to do with the DM and the campaign he or she is running, what the basic assumptions are, the type of quests, etc.
But without having read more than the first few posts of this thread and yours, what also came to mind is "adventurers" not only as low fantasy, but also lower powered, and "heroes" as not only high fantasy, but higher powered (more "gonzo"). In low fantasy, the characters are assumed to regular mortals who may be talented in this or that, but work their way up. In high fantasy, the characters often have some kind of Hidden Power, perhaps due to a Secret Destiny or being a member of a Lost Race of some kind.
In that regard, the RAW in the PHB seem to be more towards adventurers who can
become heroes. But I think this is also where an "epic/heroic" module would come in handy; what if the DM wants to run a campaign in which the PCs are akin to
ta'veren in the Wheel of Time? I could see higher ability score spreads, and some kind of "Epic Destiny" feat that gives PCs special powers.
In other words, this approach would simply be about higher level characters, but characters starting at level one with something extra special about them - akin to the protagonists in epic fantasy stories. I mean, Rand Al-Thor isn't just a high level channeler - which is one of the weak points in D&D's history in that it has never really had a mechanic for "special destiny characters" - and I think the rules should reflect that "extra special something" that epic heroes have -
if the DM and players want to play that sort of campaign.