Yes Intervals.icu keeps a copy of everything and does not need to go back to Strava or Garmin Connect once it has dowloaded an activity. However the Strava API terms of service require Intervals.icu to delete all your Strava data if you de-authorize Intervals.icu or if Strava revokes Intervals.icu API access. Strava also does not provide the original activity file via their API.
Garmin does not have these restrictions.
So Intervals.icu is a good way to externalise activities from Garmin but not so much from Strava.
You can also connect Intervals.icu to Dropbox and it can download activity files from there (for TrainerRoad and others with Dropbox support). Intervals.icu can also upload incoming activity files from Garmin and other sources to a Dropbox folder giving you an extra backup all in one place.