Against what much be a general and very broad divination? It would not take much. It is hard to tell someone's talent when you are standing right next to them. Also what the Ministry of Integrity is really looking for are wild talents that are unconnected to the empire. The sort of people likely to become bandits and rebels. If you are part of a clan able to get even the most basic warding then you generally are not what they are looking for.
We can only really speculate, since we don't know the actual mechanism the Ministry of Integrity uses to find persons of interest (after trawling through the threads the best I could find was a request from yrsillar to please stop asking lol). For all we know it could be managed by low level cultivators working from Xianmen, or Green+ cultivators individually checking every major settlement, or anything in between.
That said, I think it is unlikely that they couldn't get past low-level wards if their job is to search out potential threats. I'm reminded of someone like Gan Gaungli, whose father was a low ranking soldier who got crippled and then tossed aside. That is the sort of situation that could lead someone growing up hostile to the empire while still having access to basic warding. Also, if their biggest fear is a Sage-Emperor type situation, where a highly talented rogue cultivator gets adopted by a powerful spirit, I think to be safe they would have to assume such a spirit would protect against divination.
The Ministry only checks people in the ideal cultivating age range, so they'd be checking children and teens. Qingge was either at or past that limit by the time she ran away from home, considering she got pregnant shortly after.
Anyone older than 16-18 don't need to be checkdd. The age debuff will deal with them
Do you have a source for that? I've been looking but can't find any reference to the Ministry's only looking for people in the ideal age range. If their job is to search for potential threats a talented mortal in their late teens could easily grow to be as dangerous as someone a few years younger if they attract the attention of a powerful spirit. Particularly if enough resources can counteract the age debuff as you suggest.
Regardless, I don't think that you can say that Qingge wouldn't have been checked because she got pregnant and was therefore older than 16-18. After searching through the threads the only solid reference to Qingge's age that I could find was from Turn 8: 2-1:
"Ling Qi, I was never prepared to be a mother, at an age barely older than you are now. My conditions may have left me in a poor state, but that only explains my mistakes. It does not excuse them," Ling Qingge retorted sharply. "So do not spare my feelings."
If I'm remembering correctly Ling Qi is currently 15, and while the 'barely' gives some wiggle room I don't think that could be stretched much beyond a year or two. Adding on the time it takes to actually be pregnant and I think it likely Qingge would have been somewhere between 15 and 17 when she ran away from home, which is a decent age to start cultivating.
While it's possible that she really is as talented as Ling Qi and was missed by her clan and then passed over by the Ministry due to bad luck or something, I think the far more likely answer is that she simply wasn't talented enough to warrant their attention.
