Hello, after hours of struggling because I thought my system was poorly designed, I realized there was a very specific Minecraft bug. I provided a datapack ready to use to reproduce the bug if necessary.
The "lectern" command:
/setblock ~ ~ ~ lectern[has_book=true]{Book:{id:"minecraft:written_book",Count:1b,tag:{title:"",author:"",pages:['{"score":{"name":"@p","objective":"test"}}']}}} replace
Here, it displays the score of the nearest player, and it works fine. If you destroy and replace it in the same location, it will update the score.
But for some obscure reason, in the mcfunctions, if you destroy and then replace the block lectern , the lectern will retain the NBT of the first lectern.
As a result, I created a map that uses the mechanics of lecterns to store scores in books, but my map doesn't work because of this bug.
However, when you use the /data modify block command, it updates the chunk/data, and then it will work.

In Loop

tag @e[type=item] add check
setblock 0 0 0 lectern[has_book=true]{Book:{id:"minecraft:written_book",Count:1b,tag:{title:"",author:"",pages:['{"score":{"name":"@p","objective":"test"}}']}}} destroy
tp @e[type=item] @p
scoreboard players add @p test 1

In setup

scoreboard objectives add test dummy
forceload remove all
setworldspawn 0 0 0
tp @a 45111 100 44512121
gamerule doTileDrops false
scoreboard objectives setdisplay sidebar test

So with this code, you should have a book with your current score every tick, but you'll notice that you always have the same book...
For my personal use, I solved the issue by placing an empty lectern and then filling it with data modify...
Instead of placing it directly with the correct NBT because it doesn't work.