Personally, I'd lean into this even harder. A sorcerer isn't a spellcaster at all; they're a magical being. As they level, they get more and stronger special abilities based around their magical theme. The sorcerer "class" has almost nothing, other than granting saves/skills/hit die/ASIs.
I could also get behind a sorcerer picking one, and only one, domain, sort of like the 3.5 PrC Divine Crusader. A "domain" has exactly one spell of each level, organized around a theme (Fire, Ice, Fear, Nature, etc.). The sorcerer casts these with sorcery points. The sorcerer subclass is what grants the theme of the character (draconic, fey, celestial, etc.), and generally gives the combat scaling, utility, and ribbon powers. The domain and subclass combined gives the overall concept, like a fire domain draconic sorcerer.
So I want to follow the open-closed design principle.
In this case, I want the sorcerer to
pick spells from a broad list. They get these spells as
spell like abilities that are usable one/long rest.
Some mechanism to use them more often exists. Maybe when you cast them, you make a check - a 50-50 check, and if you succeed, you can use them again. Or maybe on a short rest, you make a check to recover spells you used in the last short rest. Or maybe you can always use a spell twice - but after the 2nd time and further, you have to check to see if you lose it.
The point of this is that whenever D&D adds a new spell, it is
easy to add it to the sorcerer's list.
For a Domain, the Domain might include a list of spells
plus spells by description. Like, a Fire Domain might include a list of spells, plus
any spell that deals fire damage, but restricted so that it only deals fire damage. So you could pick up Elemental Shield, which can be reversed to deal Cold damage; if you take Element Shield under Fire Domain you can only use it to deal Fire damage.
The ability to list specific spells exists, possibly with modifications. Like, Conjure Elemental works as a Fire Domain spell, but you can only Conjure fire elementals.
This also means that a Storm Domain could include lightning and thunder damage spell, plus a selection of cold damage spells.
And because you aren't picking Domains, a given Domain can be as broad as it needs to be without giving a sorcerer who has that domain all of the spells.
I don't want to add a whole pile of balance work on every spell. I want inclusion in a domain to be mostly a flavour thing (does it match the flavour?), not a balance thing. And leveling up your Domain is something you are expected to do.
The replacement of spell slots and points with spell like abilities, meanwhile, means that you aren't going to be spamming one ability. Each of your known spells is a separate spell like ability you get a certain number of uses of. Maybe recovery of said abilities is a common resource, like you can recover (sorc level) total SLA in a short rest once per long rest, so you aren't forced to have all of your spell picks be "use it all the time".