Well, this comes down to spell selection, doesn't it? If you want to play a "sorcerer" using the wizard class structure, you can choose blasting spells and a subclass like Evoker or War Mage that gives you innate powers of blasting. If you want to play a "wizard" using the sorcerer class structure, you can choose utility spells and any subclass that you think is cool.
oh sure you can and should be able to build your wizard to blast and your sorcerer for utility but my point was more that each uses their inherent class strengths to lean into their own area respectively.
the sorcerer's metamagic allowing them to empowered spell that fireball to burn everything to a crisp, they can throw out that powerful AoE and careful spell to protect their teammates, they can heightened spell to make sure the enemy fails their saving throw on disintergrate, metamagic is the sorcerer's tool and metamagic is mostly designed to get the most out of being used on combat spells.
meanwhile wizard's extensive spell list with a ton of niche problem solving spells and rituals, their many spell slots, their ability to learn new spells from scrolls and spellbooks seems like the perfect setup for a utility oriented 'problem solver' character, a locked door and no-one has thieves tools? don't worry i have knock! need to track down that shady NPC from a while back? locate person! we need to scale a giant cliff that's crumbling and fragile? fly! dimension door! spider climb! take your pick! (yes the wizard still has a ton of unique offensive spells but i think that's because of holdover from when they were the only arcane caster and people riot at the idea of ever taking anything away from the wizard, let wizard find their new niche)
so yes, you could build your blaster wizard and utility sorcerer, but given each their unique tools i think you would never quite reach the effectiveness of the utility wizard and blaster sorcerer and i think that's fine, lean into your strengths, do what you do best rather than trying to do what the other guy is doing best at and we'll have a funner game as a result of it.
Personally I always saw the Sorcerer as te D&D equivaalent of Marvel's mutants and DC's metahumans. Youget a suite of spells associated to your power. A Dragon sorcerer would get elemental, fear, and flight effects. A Shadow sorcerer would get necromancy and illusion spells. Sorta like domain spells. A sorcerer gets 1 domain spell list and gains new lists as they level up and evolve.
The number one thing I always though sorcerer should be able to pull magic they don't know/prepare out the Aether/Weave at a higher cost. You are a dragon sorcerer and don't have Wall of Fire prepped. Well you tap into your fire magic and cast Wall of Fire.
oh no disagreement there, getting a bunch of thematic/domain spells to your source of magic, but i don't think that contradicts what i was saying about them having the ability to channel large amounts of magic into more powerful spells and effects, however they're manifesting it they still have an innate ability to tap into the primal forces of magic and let loose due to their natural connections to it rather than the wizard who has to use tools and knowledge to manipulate it,
and the idea of every sorcerer bloodline having their own associated tag for whatever spells they can just call upon but burning higher slots is really cool,