Is there a way to get rid of activity history in Trello's cards in the same way you can delete comments for a card?
You can't delete the activity history of a specific card. However, you can effectively do this by copying the card and deleting the original card. The new card will have the same name and details as the old card (make sure to copy across any checklists and members). The new card's activity history will only show one action:
User copied this card from Card Name in list List Name
If you're using Chrome or Firefox Browser, you can hide the activity history in your Trello cards using Hide activity entries extension.