How would one put a loot table (of any kind) into a chest?

For entities, you can use:

/data modify entity @e[type=sheep,limit=1,sort=nearest] DeathLootTable set value "minecraft:entities/spider"

and the same goes for blocks:

/data modify block <x> <y> <z> LootTable set value "minecraft:chests/village/village_toolsmith"

there is a list of all loot tables on the Minecraft wiki: https://minecraft.gamepedia.com/Loot_table#List_of_loot_tables. Just expand the "File structure" section.