Harmon Kardon is one of the better sounding AV Receivers for music, but it's still a receiver not a Hi-Fi amp. However, I think the main characteristic you're hearing is the soft dome tweeter in the Missions. The sort of sound you're looking for is typically associated with metal dome tweeters instead.
Before you go spending big bucks to change components there's a couple of things you could try with the existing system.
Speaker position... are they backed up against a wall or a corner? If so, do you have the bass bungs in place? Also as a test try the speakers further out in to the room. They do like a bit of space behind them. If this improves the sound then you know it's because of position rather than a limit of the speaker's sound itself.
Have you put any weight in the bases of each speaker cabinet? There is (or used to be) a lead shot type product called Sound Bytes. Alternatively use kiln dried sand. Put a thick plastic bag in to the mass loading hole and then progressively fill to the point where the bass and midrange tighten enough. (The bag makes the sand easy to remove should you move house or sell the speakers.)
The HK AVR has optical and coax inputs and a built-in DAC. Have you thought about going out from the PC on some sort of digital output? Also try any sort of "source direct" mode off the AVR and see how that affects sound.
The final tweak is something I am not a big fan of because of the way it changes the character of a speaker's sound, but it does produce a sharpened treble response...
bi-wiring. The effect is to accentuate the bass and treble at the expense of midrange which is where voices and most instruments are.
Ultimately the M71s are probably the first candidate for a change. They are, after all, budget floorstanders. The question is what to change them for. The answer very much depends on your room, and where you have the speakers placed, and your budget of course. Changing the speakers is though a double-edged sword. New ones may well reveal the limits of the amp. Let us know how you get ion with the tweaks and tests first.
