There's a certain sweetness that comes through with a well done shot, along with a real smoothness on the palette without sacrificing body. Certainly all coffee has a bit of bitter going on, but with a properly done shot -- under real pressure, with a dialed in set of variables (grind mass, desired shot mass, shot time, etc) -- it's just part of the party, not the whole thing.
And you can absolutely get this at home, but you need a proper machine. (A Moka pot won't do it. People might LIKE Moka coffee -- lots do! -- but it's not the same thing.) I'd rather pull a shot from my Rocket than go to any shop in town. And, as I said, I've had shots just as good from machines much cheaper than my R58 (like the aforementioned Silvia).
IME, genuinely good espresso is a fairly rare thing. The set of coffeeshops that pull consistently good shots is a much, much smaller subset of [Coffeeshops] than we might expect.
What an aeropress produces is more akin to French press than anything else I've had. It's absolutely nothing like espresso. Since I'm not a French press fan, I end up not caring for it, but that's a preference thing, not an indictment of the method. When it comes to filter coffee (which is to say, non-espresso), I prefer Chemex.