But the primary reason that people multiclass is just to make a more powerful character
I would disagree with this. I think most multiclassing is to make a unique concept.
Also when talking about power that varies quite a bit by level. If you look at character level from 1-20 a multiclass character will be more powerful at some levels and less powerful at others compared to a single class character from one of their classes. This is almost always true. In 2024 5E, no multiclass combo is going to be more powerful at every single level from 2-20. I would agree they can be generally better with some planning.
For example a Bladesinger 6/Rogue 1 is a bit more powerful than a Bladesinger 7 even without the extra 4th level slot. But a Bladesinger 5/Rogue 1 is a lot less powerful than a Bladesinger 6. Similarly a Bladesinger 4/Rogue 1 and a Wizard 2/Rogue 1 are substantially less powerful than a Bladesinger 5 or a Wizard 3.
I mentioned my PC above who is currently a Elemental Eye Warlock 4/Eldritch Knight 3. This is a 1-20 campaign and eventually she will be a Warlock 4/Fighter 12 and the last 4 levels in something else, probably Bard, maybe Paladin, maybe Rogue. In any case if you just go through levels 1-7 so far and compare her to a single class Warlock:
1. F1 - Substantially worse than a single class warlock, mostly because she was a weak fighter with a relatively low Strength (15) and Dex (14)
2. F1W1 - I would call this a push. Heavy Armor, 2nd Wind and weapon mastery vs 2 spell slots and 3 invocations.
3. F1W2 - I would call this worse because a single class Warlock has a subclass here
4. F1W3 - I would still call this worse because she does not have a feat/ASI yet
5. F1W4 - Now she is better than a W5. Killer blade Cantrip, Flames of Phelethegos is online.
6. F2W4 - Better. Action Surge comes online, although it is not as good as it is on most fighters because her attacks without spells are not spectacular, Tactical Mind is online
7. F3W4 - Better. She has 1st level Fighter spell slots now so she can cast her 1st level Warlock spells without using a 2nd level slot.
So unless you are playing a one-shot, if you decide to multiclass there are times it will be really beneficial and times it will be a detriment. I would agree though that with planning it will generally be better in terms of power more often than it will be worse.