My view on the concept of multiclassing vs. any edition's mechanical representation of multiclassing need not be the same.
Concept
Imagine that the concepts that can be realized with a single class are like coins scattered on an index card. They cover a lot of it, but not all. Multiclassing allows coverage of concepts that are between coins, filling in those gaps. The system still can't handle concept that would be represented by parts of the index card not covered and having an edge.
For a system with classes as primary choice like D&D, it's basically required otherwise you are leaving a lot of character concepts unrealizable.
Realization
Go back to early D&D under TSR, and multiclassing was effectively designing a split class where you picked two or three classes at character conception, with what is allowed based on your race, and advanced in all of them equally at half or a third rate. This locked in who you were, but back then different classes needed different amounts of XP to advance levels. You got half (or a third) of your HPs when you picked up a level, etc. This locked you into a specific progression.
In AD&D 2nd (and I forget if it was also back in AD&D) was the concept of Dual Classing, which was only open to human, who were not allowed to multiclass. You could eschew your class and pick up another at 1st level. If you used any of your features from your first class you got zero XP for the current adventure. But once you were higher level in your new class then you could mix and match. This was actually closer to the 5e multiclassing, except for not being able to use anything and that it was a single allowed switch.
3.x and 5e allow picking each level, giving both a lot of flexibility in how much of each class as well as because-of-what-happened-in-game chance to change your character's direction. However, unlike TSR-era multiclassing you couldn't start with your concept. You couldn't be a FMU - Fighter/Magic-User - from the get-go.
We also have multiclassing feats as an interesting take in there, either as replacements for "true" multiclassing, or as mechanical bridges to allow iconic abilities to continue to grow with a multiclassed implementation.
Really, none of the D&D multiclassing options have hit all of the reuirements for me, and many of them have additional issues in terms of cherry picking or shooting yourself in the foot that can make mechanically a multiclassed character outside the class power curve, either too good or not good enough.
Conclusion
In a heavily class based system, a multiclassing method greatly enhances the spread of concepts that can be achieved. However, D&D's mechanical realization of that have always had both strengths and flaws and can use with improvement.