Yes, this is a HUGE issue. I beleive it may only be with iOS and not Windows. But nevertheless, it defeats the whole purpose of the Sidebar - to automatically log and track your contacts without you physically having to push the HS icon and wait 3 seconds for it to load - very cumbersome!
This should be a very easy fix. Most other add-ins on Outlook have an option to pin the app - but Hubspot does not for some reason.