Whenever you use the /tp command facing the display entity it will make you look 2 blocks above where the actual entity is
For example if you use this command

/tp @s ~ ~ ~ facing entity @e[type=block_display,distance=..5,limit=1] feet

you will not actually face the entities exact location.

But if you were to use this very long version

/execute as @s at @s anchored eyes facing entity @e[type=minecraft:block_display,distance=..5] feet run tp @s ~ ~ ~ ~ ~

you get the desired behavior

EDIT:

I'm just dumb and didn't think of checking the other entities.
Everything works as it has been facepalm

Thanks to Dhranios for making me aware of that