This leads me to ask:
Does anyone in here just say no to multiclassing?
I do the opposite, and actively encourage multiclassing at my table. I'll help to make sure all the characters are effective enough to be fun to play, but otherwise I'm fine with any combination of classes, including dips, even splits, and anything in between.
Whether or not the player wants to incorporate any of the default fiction of their character's classes I leave up to them. A fighter/monk could have been expelled from a monestary, a brawler who found a teacher for formal training, or just a character with an unusual self-taught fighting style. Similarly, I leave warlock pacts up to the player: it could be an ongoing relationship, a one-time deal, a carryover from an ancestor making a deal, unknowing siphoning of the patron's power, or anything else. (Hexblades can be as simple as the character possessing an eldritch weapon that they figure out how to use as they gain Warlock levels.)
As I see it, every character develops along their own, unique path. Sandy the Fighter/Wizard isn't, IC, erratically proceeding along two separate tracks, she's becoming increasingly powerful in her role as Sandy. That the progression happens to be granular rather than continuous doesn't bother me any more than it does for a single-class charater. (And even that isn't much--granular advancement is a necessary consequence of a level-based game.)
Why do I take this approach? I find it leads to more mechanically diverse characters at my table, which in turn reduces reliance on class identity to define who each character is. Even the players of single-class characters at my table tend to define their characters less by their class when sitting at a table of characters who can't be so easily pigeonholed.
And to address the inevitable question of why I don't choose a classless system, I find that flexible class-based systems tend to (ironically) produce greater mechanical character diversity than classless systems. In a point-buy system, the opportunity cost of taking a particular ability is generally similar for all characters. So, for example, all of the characters at the table are likely to use either the most powerful defensive ability (if they want to emphasize defense) or the most cost-efficient defensive ability (if they want to emphasize some other aspect of their character). So frequently most of the characters end up with one of two defensive abilities. By contrast, the opportunity cost of getting a particular ability in a class based system depends on how many levels a character already has in that class. This, for example, leads to the most efficient defensive ability usually being different for each character, so a greater diversity is seen in play.
