Saving throw is not an ability check the same way an attack roll is not an ability check. Ability checks are things such as "make an intelligence check to see if you remembered the code" or "roll a Wisdom (Insight) check to see if he's lying." A saving throw is a separate category of rolls that does use an ability modifier to determine the roll but that does not make it an ability check. As confirmation of this I've found the sage advice article with Jeremy Crawford confirming this.
https://www.sageadvice.eu/2014/08/23/jack-of-all-trades-bard/
https://www.sageadvice.eu/2014/08/23/jack-of-all-trades-bard/