@teacake I don’t think getting rid of trust levels is a good idea, honestly. It prevents a lot of spam and rule breaking (e.g. Users can only upload images at a Level 2, you can read the full privileges and perks of trust levels here), as well as allows us to see whose new (which can help in identifying sock puppet accounts).
Granting users these privileges right away, before they’re used to the forum and familiar with the rules would be a nightmare for those of us who mod (Also removing trust levels would affect moderation too)
Badges are just a fun little customisation for your account, so I don’t think removing them makes much sense either, it shows a record of what you’ve done/achieved on the forum.
In addition, not everyone uses the same email between forum and shopping site (others have mentioned this above), so they would have an extra hurdle of proving their identity between those accounts, which may take away the feeling of being anonymous. You’re also assuming that everyone on the forum writes reviews, which may not actually be the case.