Blue
Ravenous Bugblatter Beast of Traal
Why allow multiclass
My standard analogy of classes and multiclassing is dropping a handful of coins on an index card. Each coin is a class. Places under the coins are concepts you can build with that class. Some coins overlap: these concepts that multiple classes could build well like an archer. But there's spaces between the coins, and spaces between the coins and the edge of the card. These are concepts you can't represent well mechanically with the existing classes.
For the places between coins, multiclassing allows you to cover that space. That's a bunch of concepts that the system would not be able to represent well with a single class.
(Spaces between the coins and the edge need new mechanics, be it classes, subclasses or other character customization like feats to cover.)
Multiclassing also allows you to adapt to what's happening instead of setting your character's mechanical path at the beginning of the game and beign locked in. Say your none-too-religious rogue gets saved at great risk by your cleric and finds religion. You can roleplay your conversion, but multiclassing would let you follow up with eventually taking paladin levels.
Multiclass balance
My experience in 5e is multiclassing is fine from a balance perspective. This wasn't the case in earlier editions. For the most part, the optimizers I knew went multiclass wild when 5e came out since that was the source of power in 3.x, but now they all play single-classed characters. That's purely anecdotal, but of the tables at my FLGS that are all heavily optimized, multiclassing is definitely in the minority.
You have more flexibility but are behind in terms of power. Delays in getting new spells levels know, extra attack, etc. The 5e classes are pretty well balanced against cherry picking though some cleric domains grant proficiencies at 1st.
The caveat is that the trade offs prevent a number of combinations unless you are starting at high level. If your character wouldn't get your first ASI until 8th and extra attack at 9th, you'd be pretty fare behind for a good chunk of your adventuring life and people end up not going for it. But if you are creating characters at 11th then it's a bigger deal since you can ignore that lag that would normally act as a gateway.
Why not allow multiclass
The best arguments I've heard in 5e about not allowing multiclassing is (a) simplicity - especially with new players so they can only focus on their class and not have to even read the others, and (b) it's pretty easy to make a poor multiclass character who lags too far behind that it's not fun.
Also, if you use all of the Unearthed Arcana playtest options, they get multiclassing balance polishing later then the publish for playtest and feedback pass - they are not necessarily balanced for multiclassing.
Conclusion
I enjoy multiclassing as both a player and DM. I feel it adds flexibility in character concepts without adding in system mastery power creep. Players are free to multiclass or not as is right for them, and will not gain nor lose spotlight based on their decision.
My standard analogy of classes and multiclassing is dropping a handful of coins on an index card. Each coin is a class. Places under the coins are concepts you can build with that class. Some coins overlap: these concepts that multiple classes could build well like an archer. But there's spaces between the coins, and spaces between the coins and the edge of the card. These are concepts you can't represent well mechanically with the existing classes.
For the places between coins, multiclassing allows you to cover that space. That's a bunch of concepts that the system would not be able to represent well with a single class.
(Spaces between the coins and the edge need new mechanics, be it classes, subclasses or other character customization like feats to cover.)
Multiclassing also allows you to adapt to what's happening instead of setting your character's mechanical path at the beginning of the game and beign locked in. Say your none-too-religious rogue gets saved at great risk by your cleric and finds religion. You can roleplay your conversion, but multiclassing would let you follow up with eventually taking paladin levels.
Multiclass balance
My experience in 5e is multiclassing is fine from a balance perspective. This wasn't the case in earlier editions. For the most part, the optimizers I knew went multiclass wild when 5e came out since that was the source of power in 3.x, but now they all play single-classed characters. That's purely anecdotal, but of the tables at my FLGS that are all heavily optimized, multiclassing is definitely in the minority.
You have more flexibility but are behind in terms of power. Delays in getting new spells levels know, extra attack, etc. The 5e classes are pretty well balanced against cherry picking though some cleric domains grant proficiencies at 1st.
The caveat is that the trade offs prevent a number of combinations unless you are starting at high level. If your character wouldn't get your first ASI until 8th and extra attack at 9th, you'd be pretty fare behind for a good chunk of your adventuring life and people end up not going for it. But if you are creating characters at 11th then it's a bigger deal since you can ignore that lag that would normally act as a gateway.
Why not allow multiclass
The best arguments I've heard in 5e about not allowing multiclassing is (a) simplicity - especially with new players so they can only focus on their class and not have to even read the others, and (b) it's pretty easy to make a poor multiclass character who lags too far behind that it's not fun.
Also, if you use all of the Unearthed Arcana playtest options, they get multiclassing balance polishing later then the publish for playtest and feedback pass - they are not necessarily balanced for multiclassing.
Conclusion
I enjoy multiclassing as both a player and DM. I feel it adds flexibility in character concepts without adding in system mastery power creep. Players are free to multiclass or not as is right for them, and will not gain nor lose spotlight based on their decision.