I used to see logs between messages in conversations such as "You snoozed the conversation until tomorrow", but since I switched to the New Inbox I can't see them anymore.

If you can't see conversation events by default it means that they are set to be hidden. You can change it by clicking on the three dots button in your Inbox and select this option 👇You can also hide/show it using the keyboard shortcuts ⌘E on Mac or Ctrl+E on Windows.