Swarmkeeper
Hero
On ability scores.
PHB 173: "Each of a creature's abilities has a score, a number that defines the magnitude of that ability. An ability score is not just a measure of innate capabilities, but also encompasses a creature's training and competence in activities related to that ability."
From that we know that an ability score has two parts. The first part is the innate capability or magnitude of that ability. The 8 or 18. The second part is the ability score bonus which represents the creature's training and competence related to that ability. That game is explicitly telling us that the ability is more than the bonus.
PHB 173: "A score of 10 or 11 is the normal human average, but adventurers and many monsters are a cut above average in most abilities. A score of 18 is the highest that a person usually reaches."
Here it tells us how to read the base ability number(not the bonus). 10-11 is human average, so a 10-11 strength would not be someone who is very strong physically like @Bill Zebub's example. And it tells us that 18 is the highest that a person usually reaches, so we know that an 18 strength is very strong. Conversely, based on those two example we know that a 3 strength is very weak.
PHB 175: "Strength measures bodily power, athletic training, and the extent to which you can exert raw physical force."
Here it tells us that strength measures bodily power, so a PC with a low strength has low bodily power and ability to exert raw physical force, which is the opposite of someone with a powerful build, even if that powerful build somehow has a negative ability modifier.
It's not wrong to play the way Bill Zebub is saying, but it's also not RAW. This is a situation in which to play that way, you have to ignore what RAW is saying and opt into a different playstyle, much the same as when @HammerMan rolls to see how well his NPCs are performing a skill. This sort of playstyle is just not my cup of tea. It sounds like Bill Zebub would have a grand time playing that way and as long as people are having fun, they're doing it right.
For completeness, you forgot to include the section on pg 14 of the PHB entitled Your Character's Abilities.
Lots of "might"s, "probably"s, and "usually"s in the context of what "high" and "low" ability scores mean. Sounds like a player is free to deviate from those descriptions, if they so choose. And, worth noting, that section doesn't draw the line at where a middling score becomes either "high" or "low".