18th century people looked to the Ancient age for hints, ideas, and they re-invented the Ancient Greek Pankration sport in the form of boxing. But they left out kicks and grapples - original Pankration was very much like MMA. Calling the new sport pugilisme also suggested it is fists only.
Is there any reason why?
Another thing is, it is called a gentleman' sport. Should it be interpreted the other way around: not because it was soft and gentle, but precisely because it was raw and brutal enough to be useful for the purpose of helping soft, comfortable, sheltered gentlemen "man up"?
Are the two connected? They were less interested in real world self defense situations and more focused on punches precisely because they saw punches a better way to make soft young gentlemen "man up"?
Why would modern boxing derive from the pankration and not ancient boxing, which is attested in Homer, long before the pankration is attested in our record? Do you have a source on that? Modern boxing also has significantly more rules than ancient boxing did. Originally boxers sat (later they stood) opposite each other and traded blows until one either surrendered or fell. There was no dancing around the ring, no cuddling, none of that. The pankration also didn't feature punches much, from what we can tell. Most of our descriptions of pankration matches mention mostly grappling, with almost no punching or kicking. The pankration also rapidly devolved into a ground fight, rather than some sort of kickboxing spectacle--the fighters didn't really remain on their feet for long. So I'm a bit confused where you're getting the idea that boxing derives from the pankration, and I'm wondering what your reason to believe this is
This topic could get very very broad, as the history of unarmed "dueling" is shown on ancient pottery and wall murals, and each culture has a huge and rich history of unarmed "dueling".
I say "dueling" because "boxing" and "pugilism" are historically supposed to be an extension of "dueling" without the dramatic loss of life inherent in lethal combat. Granted, "Lethal Dueling" still existed (Burr vs Hamilton) in relatively "modern" eras.
You have specifically mentioned "Boxing", though. We Westerners (North America/Western Europe) generally trace our boxing history through London and the various rule-sets that emerged to minimize the lethality of unarmed dueling. Unarmed combative systems frequently develop in response to:
Accepted rules and regulations; this guides the development of "legal" techniques and efficient means to "win" the encounters.
Environment during development; peaceful, artistic time periods may create far different systems than war-torn periods.
Combative culture; different societies have various values and traditions. For instance, Chinese wrestling systems look and feel far different British wrestling systems. The progression of a culture can drastically alter the sort of athletic combative rules that develop.
Back to your questions:
Why does boxing primarily focus on fists?
Occam's Razor points to England's boxing rule-sets, which over the years evolved into our current sport of "boxing" and the general Western implication that "Boxing" is different than "Full Body Dueling".
Jack Broughton is a prominent English figure who heavily influenced the English boxing rule-sets. This system was further refined in the London Prize Ring Rules, then overhauled in the Queensberry Rules. Any sort of Western cultural reference to "Boxing" can likely be traced back through these history Unarmed Dueling rule-set revisions.
Why did Boxing develop as a primarily fist-focused competition? Probably due to Broughton's seventh rule:
That no person is to hit his Adversary when he is down, or seize him by the ham, the breeches, or any part below the waist. A man on his knees is to be reckoned down.
Kicking would be frowned upon because the risk of falling (and thus being considered "down") from imbalance would be too great as well as removing the temptation to kick a downed opponent. Grappling was likely discouraged because the risk to be pushed/shoved to the ground and thus "muck up" a "fair match"; also, England has a long and rich wrestling history apart from Boxing. If you really wanted to grapple, then England had other competitions and clubs for grappling. "Boxing" was where teeth were knocked out and noses bloodied.
As for Boxing being a "gentleman's sport"? The "gentle" in "gentleman" is NOT in reference to the dictionary word "gentle"; the term "gentleman" does not mean "soft" or "delicate". Instead, it derives from the word "Gentry"; basically, meaning "noble", "wealthy", or "well-bred". Being a "Gentleman" meant you were a civilized, professional, wealthy man of good breeding and respected repute. You weren't a filthy laborer, or a smelly fisherman, or a diseased beggar; you had "class", and style, and "means".
A "Gentleman" is not a thuggish murderer, but a high-class man of respect. A "Gentleman" is civilized and fights in an honorable, customary, and recognized manner. A "Gentleman" does not kick another man when they are down; nor do they attack a man from behind; nor do they attack a non-equally armed man. A "Gentleman" is a thoroughly English/Western cultural structure that influenced the rule-set for "Unarmed Dueling" or "Boxing".
Modern Boxing developed out of the English bare-knuckle boxing tradition. It was explored and refined over the 1700s by a number of figures, notably James Figg, Daniel Mendoza and Jack Broughton. Some examples of training manuals from the era;
Boxing of the era did not leave out kicking or grappling. Source #1 above states on page 22 (page marked "12") that "the cross-buttock grapple is performed when you and your antagonist close in with your right sides in contact, or alongside each other, and is gained by a low hold of the waistband of the breeches with the right hand, and of the right shoulder with the left hand, by which you capsize your opponent, by throwing him head foremost over your right hip, and part of your buttock, and is of all falls the worst." He also comments that you should "avoid the cross-buttock grapple, unless you are entirely master of the art" demonstrating that there was an expectation of spirited defense against grappling.
Jack Broughton's rules in the 1730s formed the basis of the London Prize Ring Rules of the 1800s, which still expected throws and kicking, but insists on ensuring that neither combatant has overly spiked shoes. The English county tradition of kicking even spun off into it's own lineage called Purring.