Multi-classing doesn't generalize Single-class spellcasting?

Question

I am aware of our go to question about spellcasting and multiclassing, and I am aware of the rules. I am nonetheless confused about something:

If we use the multi-class rules, in particular

You determine your available spell slots by adding together all your levels in the bard, cleric, druid, sorcerer, and wizard classes, half your levels (rounded down) in the paladin and ranger classes [...]

So, a 5th level paladin contributes as a 2nd level caster (5/2 = 2.5, rounded down 2). A 2nd level caster would have 3 first level spell slots, and that's all. However, a 5th level Paladin has a spell slot table that clearly reflects a 3rd level spellcaster. In general, it looks like the Paladin level is being rounded up when defining the class' spellcasting table. The same can be said to the Ranger and to the "1/3 casters" (Arcane Trickster and Fighter).

So, when you multiclass, the half-casters and 1/3 casters contribute less to the spellcasting levels than their own progression suggests. This means, for example, a 3rd level Paladin and 2nd level Ranger, although "jointly" count as a 5th level "half-caster" (if you sum them before dividing and rounding¹), when multi-classing the character is only a 2nd level multi-class caster and doesn't have 2nd level spell slots, even though a single-class 5th level ranger or a 5th level paladin would both have 2nd level spell slots.

This is weird to me, so, to start with, is the math right? Am I missing something? Or multiclassing is, in fact, inherently harming the spell progression provided by the not-full-spellcasters, in a way that 5 levels of different half-casters is simply worse than 5 levels of the same half-caster?

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¹ I just realized there are three answers. I only read the accepted, but the most upvoted contradicts it. Anyway, dividing and rounding before makes it even worse.

Answer 1

Your math is correct (assuming sum first, then round down).

The rule for rounding down sort of gimps the progression of multiclassing half casters. When the sum of all half caster levels is even, the caster contributes slots as expected, when it is odd we see the problem you have outlined here: it seems to be a level behind. It takes two levels in the half-caster class to contribute one level to the multiclass spell casting table, and you cannot contribute half levels.

To your specific question, is multiclassing inherently harming the spell progression provided by the not-full-spellcasters, in a way that 5 levels of different half-casters is simply worse than 5 levels of the same half-caster?

Yes.

The overall progression remains the same, but it does change arrangement of the individual level progression, delaying slot gains at every other level. You are correct to observe that 5 levels of different half-casters is worse than 5 levels of the one half-caster. But 6 levels of different half-casters contributes the same as 6 levels of one half-caster.

Here's a graph. Notice at even character levels, the total spell slots is equal, and at odd levels, the multiclassed half caster is a level behind the pur half caster.