While I need to disclaim that I haven't tried this myself yet, I would think you have three options to hide those "special" goals.
The Gentle Approach would be changing them to page events instead - and leave all boxes unchecked. Page events are really the same template as goals anyway -behind the scenes, goals are just page events with the "isgoal" value set to true.
Explanation of the options from sitecore here: https://doc.sitecore.net/sitecore_experience_platform/digital_marketing/experience_profile/configuring_the_xfile/configure_a_page_event_to_appear_in_the_experience_profile
Explanation of difference between the two types of events here:
When would you use a Page Event?
Aggressively Hide Them - If page events are still showing up where they shouldn't - then create a custom transformer to hide them from the timeline's underlying dataset completely. To do this, register your transform in App_Config/Include/ExperienceProfile/Sitecore.ExperienceProfile.Client.config (of course using a patch is preferable)
Info on creating a custom transform (in this example, they are reformatting the date - but you could do use this to suppress certain events and/or goals too.
How do you change the date format in Sitecore Experience Profile journey timeline?
Go Nuclear You could go one step back from the transform and modify the query to populate the data itself (this would be the last resort). To do this, you'd modify the ExperienceProfileContactDataSourceQueries Pipeline.
Info on the entire customization process here