Grapple
- Constrictor snake = Str save
- Crocodile = Auto-grapple and restrain on a hit
- Giant Crab = Auto-grapple on a hit
Also:
Prone
- Boar = Auto-prone on a hit
- Brown Bear = Auto-prone on a hit
- Dire Wolf = Auto-prone on a hit
- Elephant = Auto-prone on a charging hit (and it can then do a Dex save trample as a bonus action)
- Elk = Auto-prone on a charging hit
- Giant Goat = Auto-prone on a charging hit
- Mastiff = Auto-prone on a hit
- Tiger = Auto-prone on a hit made with advantage
- Warhorse = Auto-prone on a charging hit
- Wolf = Auto-prone on a hit
Restrained
Frightened
- Lion = Wis save
- Quasit = Wis save
Poisoned
- Pseudodragon = Con save
- Quasit = Auto-poisoned on a hit
Charmed
- Sprite = Auto-charmed on a hit
So out of the various creatures in the PHB that apply conditions, only
one requires a Strength save. There are at least a dozen on the above list that no longer require a Str save.