In DnDBeyond, there is a passage explaining how the music and magic connects:
In the worlds of D&D, words and music are not just vibrations of air, but vocalizations with power all their own. The bard is a master of song, speech, and the magic they contain. Bards say that the multiverse was spoken into existence, that the words of the gods gave it shape, and that echoes of these primordial Words of Creation still resound throughout the cosmos. The music of bards is an attempt to snatch and harness those echoes, subtly woven into their spells and powers.
Bards are those who can discover the magic hidden in music. Even not all performers can be called a bard:
True bards are not common in the world. Not every minstrel singing in a tavern or jester cavorting in a royal court is a bard. Discovering the magic hidden in music requires hard study and some measure of natural talent that most troubadours and jongleurs lack. It can be hard to spot the difference between these performers and true bards, though. [...] But a depth of knowledge, a level of musical skill, and a touch of magic set bards apart from their fellows.
The next section Creating a Bard gives some examples on how you become a bard, that is how you acquire the knowledge, musical skill, and magic.
Some bards acquire their magical music through extraordinary means, including the inspiration of fey or other supernatural creatures.
Did you serve an apprenticeship, studying under a master, following the more experienced bard until you were ready to strike out on your own?
Or did you attend a college where you studied bardic lore and practiced your musical magic?
Perhaps you were a young runaway or orphan, befriended by a wandering bard who became your mentor
Or you might have been a spoiled noble child tutored by a master.
Perhaps you stumbled into the clutches of a hag, making a bargain for a musical gift in addition to your life and freedom, but at what cost?
You can of course craft your own story, after all you're a bard! But first, you must understand how to use hidden magic in music. How? You tell how! You decide how you acquire: the knowledge, musical skill, and magic. The guidance suggest a more traditional approach: you learn it from someone who is/was a bard, or you realize there is magic within music because of your encounter with supernatural beings.
You can get up to 38 spells under these conditions
In your own answer you already found 3 feats that provide extra spells known with charisma, for a total of 5 spells and they are Magic Initiate (from the PHB), Fey Touched and Shadow Touched (from Tasha's Cauldron of Everything2). Other feats that grant extra spells known with charisma are Telepathic (also from Tasha's) and Gift of the Metallic Dragon from the Draconic Options UA (1 spell each). This is gives us a total of 7 spells from 5 feats.
Focusing on spells from classes, though, we should note the following for each of the four charisma spellcasters:
- Bards have at most 4 more spells known than their level
- Warlocks have at most 1 more spell known than their level
- Sorcerers have at most 1 more spell known than their level
- Paladins (assuming 20 charisma) can prepare at most 5 more spells than their level
Given this, it would be optimal to have each of these classes at one such level. The more limiting ones seem to be bards and paladins. Bards having 4 spells known over their level only at levels 10, 14 and 18, and paladins managing to get 5 spells prepared over their level at levels 3, 4, 5 and 6. Fortunately, both sorcerers and warlocks are already optimal at level 1 which allows to simply take:
- Bard 14 for 18 spells known
- Paladin 4 for 9 spells prepared (5 from charisma, 2 from half level and 2 Oath Spells1)
- Warlock 1 for 2 spells known
- Sorcerer 1 for 2 spells known
For a total of 31 spells, which with the 7 from the feats totals 38. This allows spells up to level 7 to be possible and since we are getting a bunch of bard magical secrets, some of them can even be from any class. We also get 12 cantrips (4 Bard, 4 Sorcerer, 2 Warlock, 2 Magic Initiate)
Regarding ability scores, we get 3 ASI from Bard levels and 1 from Paladin but we need to take 5 feats. With 3 feats granting +1 Charisma each, this also requires starting with 17 Charisma and 13 Strength (for paladin multiclassing).
The only way I found to make this work is by using the Custom Lineage option from Tasha's. This both allows starting with 17 Charisma (when using point buy or the standard array) as well as having one extra feat to make up for only getting 4 from classes. If this option is unavailable one feat must be given up making the total only 37 spells.
1. Although Oath Spells technically come from a subclass, all published paladin Oaths have exactly 2 oath spells at 3rd level so these should be available regardless of which one is chosen
2. This answer has had the feats area heavily revised to account for later publishings (credit to user68fd for noticing these 2 feats were published on Tasha's)
Best Answer
Charisma can mean many things
The PHB defines charisma as: (PHB 178)
Note that this definition does not mention any desire to interact with others--only that such interactions are effective when they happen.
Furthermore, charisma covers a wide range of social interactions, such as intimidation: (PHB 179)
Someone who has proficiency in intimidation has a very different kind of charisma than someone who has proficiency in deception or performance, for example.
What does this mean for your warlock?
First, your warlock can be as socially awkward as they want; it's just that when they do decide to interact with others, it's very effective.
Perhaps your warlock is very scary to others, and is perfectly aware of that fact. This means they are good at intimidating and scaring other people by deliberately playing up their mental instability, but they know that such tactics don't work in all circumstances.
Alternatively, your warlock could be very social with a certain group (other arcanists, or maybe eldritch horrors?), but not with "ordinary" people. After all, the kind of social skills needed to interact with nobles is probably very different from those needed to interact with demons.
Or, it could be that your warlock is truly mentally unstable and socially awkward in all cases, but something about his patronage with the Old One makes him supernaturally compelling to other mortals. They're already getting all sorts of other weird, eldritch powers, after all.