Contrariwise, when the party starts out as a band of nobodies and then is capable of taking on a small dragon in the span of a couple of months in game, it sucks all my suspension of disbelief right out. So if you're leveling ever couple of sessions (as a lot of WotC campaigns do), either your characters are growing at a video game/anime rate of speed or your campaign is racing through time to compensate (i.e. moving in fast forward). Neither of those appeal to me at all. I've never gotten attached to a character in 3E and later editions like I did back in 2E when I might be playing the same character for a year or more.
"Improvement" came from forging a reputation, making contacts, relationships. It came from the stories we told through the sessions of play. And yes, greater capabilities as we grew in level as well. But it's not like there was nothing to the game except level increases. If that's all D&D was, I'd go play a dungeon crawler. Less scheduling conflicts.
Yes, but at least early on, you don't make much of those (reputation, contacts, stories). Thinking of my longest running character, the stories I have of levels 1-5 are:
*Getting mauled by a bear.
*Getting mauled by a shark.
*Getting backstabbed by my own party member.
It's that last one that was a turning point, and set him on a new path, but ultimately, the only thing that allowed for even those feats were several levels of gained hit points. I'm not here to say I think my way of running games is superior; but ultimately how long the process of going from zero to hero should take is entirely subjective; there's no timeline that says "you must spend X months to reach level Y", since level is a completely abstract concept.
You can't look at a decorated soldier in our world and say "ah, he's obviously level 7" and equate his age to how long it took him to reach his current status; if anything, his abilities may have degraded from disuse or lingering wounds, things that D&D characters don't typically have to worry about.
Fiction and mythology are full of many fantastic warriors who accomplish great feats at an early age; perhaps spending several months in a high risk environment dealing with lethal foes would force you to get skilled or perish.
It depends entirely on the story you want to tell. I have a character I've played on and off since about 1991, and still occasionally do to this day. In the campaign he's in, xp and gold and even most magical treasure have long since ceased to have any real meaning.
I had a character in 4e that got to level 22 inside of a year, fending off dragons and githyanki and demigods. I don't see the story of one as being superior to the other. I've been in games where three months can go by without seeing a level or a magic item. I've been in games where you get about half a level a session.
I don't see one as superior to the other; in fact, in these days, I've noticed you might as well start at level 3-5 and hand out xp like candy, because if a game lasts more than 6-7 months, that's incredible. Real life ends games way faster than running out of levels, or even getting to the levels where the DM might toss in the towel because the characters are difficult to challenge (the last time that happened to me was after 2 years, and in all honesty, it wasn't that I couldn't challenge the group, it's more that I didn't have the mental energy at that time to continue).