I'm thinking medicine as a major joint project and let me paint one scenario. Okay, well before Jenner and cowpox, let's say people circulated smallpox scabs earlier than is documented by writing, which is possible. The problem is that the potency varies widely. It may be too little for a good vaccine, or it may be too much. The beauty of cowpox is that it has a medium virulence. Please remember, Rome does have a medical tradition with Galen living from c. 130 AD - c. 210 AD. So, let's say after the date traditionally given for the fall of the Western Roman Empire of 476, let's say a Pagan doctor notices and announces to challenges that cowpox seems to make people immune to smallpox. And then it's a Christian lady at another medical who figures out the first couple of steps to test and develop this, and a heck of a lot more ethically than Jenner did. And the Pagan doctor figures out the next couple of steps. And this mix of competition and collaboration between at least two different research centers often really advances a technological field.
And all microbiology is, is grinding lenses and growing things in petri dishes. And it's building up an intellectual framework and noticing such things as, hey, wait a minute, these contaminated dishes where common molds seem to inhibit bacterial growth, this is potentially highly, highly useful.
And maybe as early as 675, Muslim physicians get into the act and do some of the middle work in understanding antibiotic resistance. And it becomes a culture norm that once a person starts an antibiotic, you really need to complete the entire week or ten days. And maybe as early as 800, ideas about probiotics are developed, if no other reason than neutral bacteria take up available real estate and take infection less likely. And taking yogurt while you take an antibiotic almost becomes a matter of common sense. And with Europe more open to Islamic ideas, and Islamic science and math, the ideas of the Renaissance both come earlier and in a sense are less needed.
The centuries later, on an alternate history website, someone writes WI: antibiotics did not exist at the time of the black plague? And someone else writes, this is almost complete ASB since by the mid-1300s antibiotics had already existed for more than 500 years!
Without people dying from unknown causes, the world is a less boogie oogie, heebie jeebie scary place. And it's less common for a person to have the heartbreak of losing a child or even losing multiple children. Plenty of diseases and syndromes still have unknown causes. But it's like there's a big group which have known causes and a big group with unknown causes, and a there's a feeling or progress being made. People still have religious beliefs, but fewer people are obsessively religious where that's all they know about.
In the OTL, progress in such things as steam engines and railroads came before medicine. It most probably did not have to be this way.
This is the scenario in which medicine saves the world, both literally and figuratively.