In official 5e rules, saving throws are not ability checks, skills are. It's not about what seems logical, those are the rules. I know that saving throws are governed by abilities, but they are not simple ability checks.

So, skills are ability checks, initiative is an ability(dex) check, but attack rolls and saving throws are their own thing. In exhaustion rules, one level of exhaustion gives you disadvantage on ability checks, but 3 levels of exhaustion give you disadvantage on saving throws and attack rolls.

So either their system uses a different hex, that affects saving throws, or for some reason either shove has the wrong wording, and uses opposing athletics check, that has wrongly written as saving throws.