I like the "modern" approach, but I wish they had different names than the abilities. Having them be the same is one of those things that seemed beginner friendly to some WotC designer's gut sense of what makes things accessible, but actually just causes endless confusion at table and requires extra verbiage in manuals. I'm not sure I've ever been in a 5e session where someone didn't confuse ability check and saving throw.
I also think the lack of separate names for the saves derived from various abilities caused the conceptualization of what the respective saves really mean to be a bit weak.
But the basic "proficiency, based on starting class, in one common and one rare save, with advances based on proficiency bonus in 5e is a nice mechanic, it just needs less clunky terminology. I know some people advocate for just dropping the rare saves, but they really don't hurt anything and give characters whose primary stat is associated with a rare save occasional chances to shine.