Unfortunatly, you can't export timeline data/engagements from HubSpot currently.
There are third party services that can access this information via the API, usually to migrate them to another system, so this sort of approach might be possible for what you are trying to do too.
I'd recommend voting and add your feedback on this idea, even though the status is currently "Not in planning" this doesn't mean that could change in the future. If you follow the idea you will receive notifications if the status changes or if there is an update from our Product team.
Would very much like to export all communication with a contact (including call logs and attachments) - Why isn't this available? Emails are available. Why not notes and call logs?
Unfortunatly, you can't export timeline data/engagements from HubSpot currently.
There are third party services that can access this information via the API, usually to migrate them to another system, so this sort of approach might be possible for what you are trying to do too.
Using a 3rd party plugin to export call/email data is NOT an accepted solution. HubSpot should STOP farming out responsibilities that should be an inherent part of the solution that WE PAY BIG $$ for.