That's a great point.
The flipside of the coin, though, is that plenty of times an injury like this
doesn't lead to anything serious.
I've been around for a while, and had tons of wounds, burns, etc., left a great portion of them totally untreated, and never had anything serious happen with any of them. Also, think how seldom you've heard of another person you know having a serious problem with a cut, burn, splinter, etc.
Preppers are fond of saying things like what you've said above. And you're right that we'll have problems without antibiotics, and with a lot of medical treatments unavailable. But as far as minor injuries that could lead to infection, maybe we're leaps and bounds ahead of people from centuries past simply by having the germ theory of disease, and can avoid almost all problems like you experienced, just by promptly cleaning, disinfecting, and bandaging the wound-- even without antibiotics.
When medical disinfectants are no longer available, they might still be improvised from beverage alcohol or maybe mouthwash, or alcohol may be homemade.
We sometimes sound as if we think losing a lot of modern tech will destroy humanity. But just think, humanity made it this far, without having antibiotics and most of the rest of of our remedies, through most of human pre-history and history. And that's even though humans used to live much more unsanitary and rough-and-tumble lives.
The number of men and women who lived before antibiotics, but who never were killed off or suffered an amputation due to a cut or scrape must be very, very many. And as I've said, their lives were more unsanitary, a lot closer to agriculture and to the land, etc. And most of the history of humanity was before the germ theory of disease.
Covering wounds, pouring wine or urine on them, and other treatments that are valid for helping to prevent infection existed before the germ theory of disease, though.