After upgrading to a basic premium account, going to my shop page and attempting to deposit an item into the inventory yields the following error message:

POOLED STORAGE DEPOSIT NOT ALLOWED
You must have a private bot if you want to deposit items. Please consider obtaining a VIP account.

Am I doing something incorrect? As far as I understand, depositing items is allowed, just into the "public" bot inventory?

That is just standard warning for which basic premium account can also be affected if Steam bans the bot that you share with another user that do fishy trades.