The Bug:
Villagers' "CanPickUpLoot" tag cannot be set to zero.
Steps to Reproduce:
Summon a villager with its "CanPickUpLoot" tag set to "0b" by using the command provided below.
/summon minecraft:villager ~ ~ ~ {CanPickUpLoot:0b}
Inspect the value of the "CanPickUpLoot" tag by using the command provided below.
/data get entity @e[type=minecraft:villager,limit=1,sort=nearest] CanPickUpLoot
Take note as to whether or not villagers' "CanPickUpLoot" tag can be set to zero.
Observed Behavior:
Villagers' "CanPickUpLoot" tag cannot be set to zero.
Expected Behavior:
Villagers' "CanPickUpLoot" tag would be able to be set to zero.
Linked issues
relates to
Attachments
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Regardless of mobGriefing, Villager's CanPickUpLoot is always true and cannot be set to false.
If mobGriefing is false, Villagers cannot pick up bread because the gamerule precedes their CanPickUpLoot.

Confirmed.
Confirmed in 1.8.8 and the newest 1.9 snapshot 15w42a
to disable villagers from taking your entity's do mobgriefing of
i tink that they have to make a other gamerule for villagers entities stealing and farming becaus when i set mobgriefing false i want no explosions no enderman but i want farmers and food giving to villagers allowed

20w06a affected

In 1.16.1 and 20w28a
Can confirm in 20w51a.
Can confirm in 21w03a.
Can confirm in 21w05b.
Can confirm in 21w11a.

Can confirm in 1.17.1.
Can confirm this behavior in 21w41a. Here are some extra details regarding this problem.
The Bug:
Villagers' "CanPickUpLoot" tag cannot be set to 0.
Steps to Reproduce:
Summon a villager with its "CanPickUpLoot" tag set to "0b".
/summon minecraft:villager ~ ~ ~ {CanPickUpLoot:0b}
Inspect the value of the "CanPickUpLoot" tag by using the "data" command.
/data get entity @e[type=minecraft:villager,limit=1,sort=nearest] CanPickUpLoot
Take note as to whether or not villagers' "CanPickUpLoot" tag can be set to 0.
Observed Behavior:
Villagers' "CanPickUpLoot" tag cannot be set to 0.
Expected Behavior:
Villagers' "CanPickUpLoot" tag would be able to be set to 0.
Can confirm in 1.18.2.
Can confirm in 1.19.
Do you have mobGriefing set to false?