Proficiency is training. Ability scores are natural ability. Refer to pg 239 of the DMG for reference.
Proficiency applies from training and practice.
No, proficiency is
additional training, as I said. Here is the passage from the PHB, p. 171. It clearly states ability scores encompass a "creature's
training and competence" in activities related to that ability.
"
Ability score is not just a measure of innate capabilities," means it is
NOT "natural ability" as you seem to think. That is what ability scores more represented in earlier editions, but not in 5E. I'll admit the change in philosophy annoyed me at first, but I've accepted it as the intent in 5E.
So, not everyone needs to be proficient in Athletics to grapple or jump effectively, as they can do it with a relatively high STR score. But, will someone who is proficient be better than someone who isn't given the same ability score? Certainly.
I think part of the problem is also the terminology. In prior editions, there were skills you could not attempt without training in them, 5E did away with that. One way I like to explain this is that it is backwards from prior thinking. The training and competence increases ability, not the other way around. Someone
becomes stronger because they are physical active and capable.
Halfling base STR and CON are not 20's. It would be a 19th level halfling body builder to approach that.
A lot of assumptions in that statement.
First, your level required is
way off. It could be a 4th-level character, to have a 20, or a 6th-level fighter to have both 20's. Remember, default for 5E is rolling 4d6k3 for ability scores. The standard array and point-buy systems are options. So, rolling two 18's (not likely, but certainly possible) and then apply your 4th and 6th level ASIs. Done, and done.
Second is the idea that this STR 20 halfling must be a body builder. What is Strength in 5E?
(Oh, look... "training" again...

)
Having a STR 20 could be muscular, of course, but much of STR is the ability to apply leverage and power. Take power-lifting for example. While muscle is important, technique is
huge, and STR measures that as well as muscle.
Regardless of how you look at it, the small creature (neither with great size nor mass) can have those high STR and CON scores you seem to think are only linked to size and mass.