A review of strength (athletics) is indeed a verification of strength – the clue stands in the word "strength".
When you initiate a gripper, you use a Strength (Athletics) exam, and if you have an advantage (disadvantage) in strength checks, you have an advantage (disadvantage) because, as mentioned above, it is a strength Exam.
If you try to escape or escape a grapple, you can use strength (athletics) or skill (acrobatics). The first is a strength check, the second is not – it's a skill check. By the way, initiative is a skill check, so the benefit (disadvantage) of skill testing is the same for initiative roles.
If you want to build a grappler you should read the Grappler Guide. One of the most important things you seek is access to Enlarge Spell, in part, to gain the benefit you determine, but also to capture huge creatures.