I'm not saying this to downplay the effects of Covid/Omicron, but rather to make the point that long-term damage from "mild" infections is not new. What matters is how many people are affected. Omicron will undoubtedly ruin the lives of some people who recover from the initial infection.
The criteria I would use is if the medical community is recommending testing for damage after illness. I an pretty use the recommendation is not to test the lungs, kidneys, heart, and brain after an influenza infection for people with no other symptoms. I could be wrong, so feel free to correct me here.
I do know however that dive doctors recommend divers to test for lung damage after a confirmed covid infection, even with no symptoms after infection. There is no such recommendation for influenza, so I would hazard a guess that the risk for otherwise healthy divers after an influenza infection is less than of covid. I have yet see a recommendation for omicron, but for now I would be cautious until the data is in.
For covid it is good to keeping an eye on oxygen levels. There are not many good tools to do this, through the skin do turn blue when things start to go very bad. Oxygen capacity can go down a fair amount before there is obvious signs, especially if the patient is laying down. The recommended medicine is pure oxygen which will require a hospital visit.
Personally I am more worried about covid. It is easier to monitor the fever of the flu, the signs for trouble is more clear, and the medical guidelines for when to go to the hospital is well established. Worrying oxygen levels are silent, harder to measure, and the medicine isn't something that people usually have at home. The guidelines are not fully established and there is ongoing research studies on the effects.
I've had a couple different SPO2 meters for a few years, they seem to work pretty reliably.
Not yet as accessible as a thermometer if you're talking about population measures, but since it sounds like this is something that concerns you personally it might be worth getting one.
What we do know about covid is for example fatigue and long term lung damage, both which can have very debilitating effects on health, wealth and social well being. Lung damage in particular can be hard to detect without doing an actually study after the infection.