First I'll showcase my "fixes".
Natural Casters
Sorcerers gain their magic not through study or discipline, but through natural, inborn magical talent. Their magic is in their blood, and so they do not require material components, or focuses, to cast their spells unless the material component has a cost listed in the spell description.
Metamagic
At 3rd level, you gain the ability to twist your spells to suit your needs. You gain all of the following Metamagic options. (this is meant to replace the PHB text, in other words, you get
all metamagic options.
Wild Mage
The wild mage has minor alterations to both its Wild Magic Surge and Tides of Chaos abilities.
Wild Magic Surge
Starting when you choose this origin at 1st level, your spellcasting can unleash surges of untamed magic. Immediately after you cast a sorcerer spell of 1st level or higher, roll a d20. If you roll a 1, roll on the Wild Magic Surge table to create a random magical effect.
Tides of Chaos
Starting at 1st level, you can manipulate the forces of chance and chaos to gain advantage on one attack roll, ability check, or saving throw. Once you do so, you must either have a Wild Magic Surge or finish a long rest before you can use this feature again.
My reasoning for the above changes is as follows:
Natural Casters. This one should be obvious. Components are ignored by a lot of DM's which I find annoying but also figure why not make it official for the class that is supposed to be an innate caster? It doesn't have a huge impact and adds a lot of flavour to the class even if DM's aren't that strict on components.
Metamagic. At first blush a lot of people might think this is OP but I think on reviewing the reality of metamagic, it's just fine. Due to the current restrictions on metamagic, almost everyone ends up choosing Quicken and Twinned and maybe Heightened or Extended later on. That's just plain boring. Quicken & Twinned are the obviously superior choices to the point where choosing anything else makes your already lackluster class choice significantly less powerful and interesting. And the thing is, even if you allow
all the metamagic options, most combats end up having the player choose Quicken & Twinned anyway. So for the rare moments where Subtle or Extended could be very cool and fun, at least they're
there rather than the player wishing they were there but lamenting having to choose the vastly superior options over them. That change alone makes the class
far more interesting and fun to play.
Wild Magic Surge. Leaving it up to the DM is just stupid IMO. DM's forget or think it's too powerful or disruptive (it really isn't), or they do it every cast anyway because they think it's fun. There is really no reason to limit it so severely or put it in the hands of the DM. Take that off their plate and just have the blanket rule of a 5% chance whenever you cast a spell rather than a cantrip. I don't like a lot of people's decision to make it more likely per spell level, as then it becomes too significant a chance of occurring to the point of ridiculousness. Spellcasts, even at higher levels, aren't so frequent that 5% chance will be disruptive but are frequent enough that that 5% chance is a constant "threat", and thus adds a constant element of "what if" to every spellcast. Players both become complacent because it's always present and hardly ever happens, but also surprised and think, "Oh crap!" when it does. IMO, 5% per spell cast is the perfect balance.
Tides of Chaos. This has to be paired with the above wild magic surge change otherwise it's not that impressive. At 5% per spellcast, it's enough of a cool factor that it's both fun and doesn't become OP from being too frequent.
These changes, IMO, bring the Wild Mage into line with the Draconic sorcerer's power. I don't see any need or reason to change the Draconic sorcerer at all. The Storm Sorcerer I think needs a tweak but currently I haven't put much thought into it and haven't found any fix I'm satisfied with or like.
Now on to the OP's quoted post:
So here's how I would fix the Sorcerer, as a base class:
Increase spell slots per day by 1 across the board. Let Sorcerers benefit from extra spells AND metamagic, rather than having to choose one or the other.
Broaden the spell list to include all Wizard spells AT THE LEAST.
A mechanic to learn spells from other lists. This makes more sense for the Sorcerer than for the Bard, to be honest. If the Sorcerer is going to be relegated to sticking to a theme, let them at least stick to it.
More metamagic. More interesting and more WEIRD. Change damage types. Cast Self-only spells on others. Transfer concentration. I don't know; but this is the Sorcerer's true defining trait and niche, it should be extremely creative.
Spell slots per day does not need to be increased, IMO. That's what spell points are for. I strongly disagree that giving them more spell slots is a good idea as I think that would most definitely tip them over into the grossly overpowered category. I really don't think it's needed or warranted.
Giving spells from other lists is just really, really bad design for the sorcerer. Comparing the sorcerer to the bard is also a false equivalency. If that was a design choice that was preferred, then it would require a wholesale rewrite of the sorcerer from the ground up to accommodate it and isn't something that should be just shoehorned into it.
More metamagic is... hmm... problematic. The scope for powercreep there is really too broad IMO. I think giving them carte blanche access to metamagic is the better option rather than bloating out the metamagic choices. I could see maybe one or two versatility options being added, but other than that, I'd be extremely wary of any extra metamagic options being added, with or without my changes to metamagic.