I was thinking that since Dominaria is swimming in oil and Mirrodin was already proto-phyrexian in a way that Phyrexia's contagion is actually really mild and everyone is paranoid for no reason.

I think everyone is super paranoid for no reason. Phyrexian Oil is not a viral zombie plague where even the slightest contamination will lead to total conversion. The reason it was so virulent on Mirrodin is because of the nature of the plane and its inhabitants. The Micosynth helped it to metastasize through the artificial world where it likely can't do as much on a more natural world. See Dominaria as an example of a natural world that is arguably steeped in the mass graveyards of Phyrexian Invasion-era oily parts and yet it seems to have pretty much moved on from Phyrexian-based problems to more traditional ones.

I am willing to bet that on a more natural world, converts need to be subjected to more of a traditional "Completion" process, where parts are removed and replaced and oil needs to be transfused or applied in a more methodical way than just "Ew, you got some conversion juice on you! I guess you're going to Borg out now."

Phyrexian oil is still a problem on Dominaria, a lot of the black cards from Time Spiral had depictions of people afflicted with it. Where Dominaria doesn't have the same problem is that Orim created a vaccine way back in Invasion to start with, so there's just treatments if nobody is that far gone.
And people probably have a resistance to it anyway.

Ah, so you do need to take your meds and it gets fixed.

The way I understand it, the problem is what happens when someone DOES fall. So the oil isn't super-contagious, it doesn't compleat or terraform in a reliable way. It kinda spreads some, and those who get it inside them in a bad way might get sick, and those who get sick might die or compleat. Barin and Azure basically cover it; Mirrodin was built to fail.

The problem is less that Phyrexia is impossibly contagious and more that it is impossibly relentless. Dominaria, which went through a lot of hell, has learned to contain the problem, has specialized tools for combating the problem, and despite concerted efforts can't eradicate it. What about J. Random Plane, someplace that doesn't know the danger that Phyrexia ultimately represents? The oil on its own is a fairly low threat to any given individual, but once higher forms emerge (or, more likely, somebody does hit the low low chance of being compleated or at least turned into an insane cultist by the oil coursing through their body), the spread then has an advocate. If that's even needed. Look at Steady Progress and Instill Infection. That's the danger; ignorance and curiosity alike advance Phyrexia's position in early phases. Sooner or later the track progresses, and the farther it goes the faster it goes and the harder it is to beat back to a previous state. Phyrexia doesn't grow from a drop of oil overnight, but it can grow or grow back from a drop of oil. In a sense, Dominaria is the best case scenario: jumping on it in early stages for the rest of time, because it's going to be extremely hard to actually destroy all viable contagion.

Ethan Fleischer and Kelly Diggs did a podcast on the ideas creative ways working with at the time of Dominaria (so soft canon that might be ignored later but what we can take for now). IIRC they said the legacy weapon neutralized the oil on the plane so no oil from the past is there to rebuild Phyrexia.

I mean, it's also worth keeping in mind that the oil on New Phyrexia originally wasn't supposed to be the regular Glistening Oil we saw in earlier stories, but rather an unfinished weapon that Karn and Jeska accidentally brought to Mirrodin. But then they retconned everything to smithereens and simply decided it was now regular Glistening Oil that had leaked from Xantcha's heartstone. Which is dumb because 1.) it poops all over Xantcha's legacy, 2.) the stone wasn't even hollow, 3.) Urza specifically stated that the stone contained no traces of Phyrexia anymore when he put it into Karn, and we're talking about an Oldwalker with X-ray eyes who could smell Glistening Oil and had spent millenia fighting Phyrexia. Since the version where the oil is an unfinished Phyrexian weapon is the only one that makes any damn sense, I just choose to ignore the retcon and go with the original canon. Heck, they didn't even remember that the heartstone is in Karn's head rather than his chest.

Amen brother.
The retcons they do for the returns they see