The restriction only applies to bonus-action spells
The relevant rule regarding bonus-action spells appears in the rules on spellcasting in the basic rules or PHB (p. 202):
A spell cast with a bonus action is especially swift. You must use a bonus action on your turn to cast the spell, provided that you haven't already taken a bonus action this turn. You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.
There are no other restrictions on casting multiple spells in the same turn. If you cast a spell as your action, it doesn't restrict your ability to cast a reaction spell. (It can, however, prevent you from casting any spell as a bonus action; the rule is agnostic of order, so if you cast any non-cantrip spell as an action, then you can't cast a spell as a bonus action.)
As you point out, the sorcerer does have the Quickened Spell metamagic option:
When you cast a spell that has a casting time of 1 action, you can spend 2 sorcery points to change the casting time to 1 bonus action for this casting.
This gives the sorcerer versatility, freeing up their action to spend their action doing other things for the cost of 2 sorcery points. However, it does not ignore the bonus-action spell rule; like all other characters, if the sorcerer casts a bonus-action spell (even if it's a Quickened spell that normally has a casting time of 1 action), the only other spell they can cast on their turn is a cantrip with a casting time of 1 action.
Although a sorcerer can't cast a quickened fireball and another leveled spell on the same turn, Quickened Spell can still be quite useful; it frees up their action to Disengage (e.g. if they want to move away from an enemy from being attacked), Dodge (e.g. if they want to minimize the chance of being hit by an attack or take a lot of damage from an enemy fireball), attack with a weapon, drink a healing potion, or anything else they can do as an action.
In fact, you could cast up to 3 leveled spells during your turn, but only if you have the fighter's Action Surge feature:
Starting at 2nd level, you can push yourself beyond your normal limits for a moment. On your turn, you can take one additional action on top of your regular action and a possible bonus action.
Once you use this feature, you must finish a short or long rest before you can use it again. Starting at 17th level, you can use it twice before a rest, but only once on the same turn.
(You could get the feature through multiclassing between fighter and any caster class, or using the Eldritch Knight fighter subclass.)
You could cast a leveled spell as your action, then use Action Surge to get another action on the same turn, and use that second action to cast another leveled spell. Finally, if the relevant trigger occurs, you could cast a reaction spell (e.g. you would get hit by an opportunity attack so you cast shield, or you get damaged by the attack so you cast hellish rebuke, or an enemy tries to counterspell your action spell so you cast counterspell on it).
In short, the rules may seem a little weird/arbitrary, but they're only interested in whether or not you've cast a bonus-action spell:
- If you have cast a bonus-action spell, the only other spell you can
cast that turn is a cantrip with a casting time of 1 action (no
reaction spells on that same turn).
- If you haven't cast a bonus-action spell, then you are free to cast a
reaction spell on your turn.