Mining modification help

I'm not exactly sure, but I think you need 4 different mcfunction files for the 4 different directions, which all break blocks in different locations as I pointed out in my last comment.

I hope I'm understanding you correctly. If that isn't what you want achieve or you came across additional questions, feel free to ask, but here's what I think you should do:

I assume, that the location ~ ~ ~ is where the original block is broken by the player. (Correct?)
First you "would like it to break only the block below", so we remove everything except the line for below:

execute if block ~1 ~1 ~ #minecraft:break_list run setblock ~1 ~1 ~ air destroy
execute if block ~1 ~ ~ #minecraft:break_list run setblock ~1 ~ ~ air destroy
execute if block ~1 ~-1 ~ #minecraft:break_list run setblock ~1 ~-1 ~ air destroy
execute if block ~-1 ~1 ~ #minecraft:break_list run setblock ~-1 ~1 ~ air destroy
execute if block ~-1 ~ ~ #minecraft:break_list run setblock ~-1 ~ ~ air destroy
execute if block ~-1 ~-1 ~ #minecraft:break_list run setblock ~-1 ~-1 ~ air destroy
execute if block ~ ~1 ~ #minecraft:break_list run setblock ~ ~1 ~ air destroy
execute if block ~ ~-1 ~ #minecraft:break_list run setblock ~ ~-1 ~ air destroy

Then you want it to "mine 1 more depth". I'm not sure whether you mean 1 more block down (1 block deeper) or 1 block more depth in the direction the player is facing, while breaking the original block (1 block more depth).
The first variant would mean you just have to add 1 line, which breaks an additional block 1 block deeper:

execute if block ~ ~-1 ~ #minecraft:break_list run setblock ~ ~-1 ~ air destroy
execute if block ~ ~-2 ~ #minecraft:break_list run setblock ~ ~-2 ~ air destroy

The second variant would mean you'd have to add two lines, that break additional blocks in the facing direction instead. Therefore you'd have to add 1 to the axis that was just ~ in all the commands you had before (for that specific direction):

execute if block ~ ~-1 ~ #minecraft:break_list run setblock ~ ~-1 ~ air destroy
execute if block ~ ~-1 ~1 #minecraft:break_list run setblock ~ ~-1 ~1 air destroy
execute if block ~ ~ ~1 #minecraft:break_list run setblock ~ ~ ~1 air destroy

This might break additional blocks, where you stand, instead of behind the block that was broken, then you'd have to replace all the 1 with -1.
Then you'd need to repeat this "for the 4 directions, as well as up and down". I assume you have a seperate *.mcfunction file for every direction.
Also, I don't know what your block tag #minecraft:break_list contains. If you just want it to break the stuff a standard pickaxe is used to break, there already is an existing block tag called #minecraft:mineable/pickaxe, so if that's what you need you wouldn't have to add a custom block tag.