Putting the ASIs into the backgrounds is very similar to putting the ASIs in the races, and there is a reason we went away from that.
True. Before
Tasha's Cauldron of Everything, races had fixed ASIs that tended to favor only certain classes. A Dragonborn character started the game with a +2 STR and a +1 CHA. This arrangement favored STR-based Fighters, Barbarians, Bards, Sorcerers and Warlocks. It also really favored Paladins who depended on both STR and CHA. But if you wanted to be Dragonborn who wanted to be a Rogue or a Wizard, you were kind of out of luck until you could grab an ASI at 4th level for DEX or INT.
Tasha's Cauldron of Everything gave everyone the chance to assign the +2/+1 or the +1/+1/+1 ASIs to any ability score they wanted their character to have. Which IMO was an improvement. As ASIs were now another element to customize your character with.
And now 2024 D&D has decided to follow in the footsteps of PF2 and Level Up by putting ASIs into Backgrounds instead of the races. However, certain backgrounds do favor certain character classes. Anyone who has followed the Quick Build rules in 5e when designing their characters knows this. You want your character to be a Fighter, the class's Quick Build suggests that you take up the Soldier background. However, you don't have to take up this background for your Fighter if you don't want to. There probably are a number of players who take up the suggested background simply because it saves them the time and effort. But there probably are a number of other players who take up another background just to put in an unexpected twist to their character's backstory.

The street urchin who became the fighter, or the acolyte who became a fighter, etc. Neither approach is wrong when developing a backstory IMO.
PF2 has it where you receive a fixed ASI of +2 from one of two ability scores (such as INT or WIS) and a floating ASI of +2 that you can assign to another ability score of your choice. This is an addition to your ancestral ASIs, your class ASI, and 4 floating ASIs at 1st level. Level Up otoh has it where you received a fixed ASI of +1 for just a single ability score and a floating ASI +1 that you can assign to another ability score of your choice. So, there is some choice here on where you want the ASIs to go when developing your character, just not a lot.
The background ASIs for 2024 D&D depend on whether or not you pick the +2/+1 option or the +1/+1/+1 option. But you can only apply these ASIs to three fixed ability scores, and these scores might favor particular classes much like how the fixed racial ASIs in the original 5e PHB did.
Since 2024 D&D is backwards compatible with 2014 D&D, it should be interesting to see which version of 5e's backgrounds will be picked up by the players. I'll be continuing with the backgrounds from 2014 D&D because there are no plans for my group to make the switch to 2024 D&D any time soon.