So, I don't mean to be a jerk about this, but it seems like you glossed over a lot of rolls that had potential botches. Every roll for reciting a ring is a potential botch roll. And it seems that over 60 years you probably recited them an awful lot. Enough times to make the chance of a botch statistically significant.
Let's look at the numbers with your highest Ars Notoria value for a reference point. The value for AN would obviously be less throughout much of Poenitens' career, but we'll start here as a baseline.
Each ring has an EF of 3 + 3*ring + 1/season; and you are guaranteed to roll an 19, unless you botch (20 with ring 7). So, assuming you went with the guaranteed success, you'd aim for an 19 EF. That gives us the following:
Ring 1: You can get this to work for 13 seasons, or a little over 3 years at at time without any trouble.
Ring 2: You can get this to work for 10 seasons, 2-1/2 years, at a time without fail.
Ring 3: You can get this to work for 7 seasons, nearly 2 years, at a time without fail.
Ring 4: You can get this to work for 4 seasons, 1 year, at a time without fail.
Ring 5: You can get this to work for 1 season at a time without fail.
Ring 6: You can get this to work for 1 season 70% of the time.
Ring 7: you can get this to work for 2 seasons 40% of the time.
Granted, once you learn the 7th ring you can recite the other rings without botch, assuming that you successfully recite the 7th ring. But that still leaves a lot of time before then when any recitation of a ring would come with the danger of a botch.
You went 45 years with the 1st, 2nd, and 3rd rings known. That's 180 seasons. You note that he's generally under the effect of all 5 rings once you know 5 and all 6 when he knows 6, so I'm assuming that's the case for when he knew lesser rings. That's 14 recitations of the 1st ring, 18 recitations of the 2nd ring, and 26 recitations of the 3rd ring.
He went 30 years with the 4th and 5th rings known. That's 120 seasons. That's 30 recitations of the 4th ring, and 120 recitations of the 5th ring.
He went 15 years with the 6th ring known. That's 60 seasons, and 60 recitations of the 6th ring.
Once he gets to know the 7th ring, things get a lot easier, since then he only has to worry about that one. But even then, it's not guaranteed. By my calculations, Poenitens has a +20 with the 7th ring [5 (Int) + 9 (Ars Notoria) + 1 (specialzation in 7th ring) + 3 (major flaw) + 1 (minor flaw) + 1 (chaste) = 20]. With an EF of 26 to make it last 2 seasons, that's a 40% chance of success. You say he'll try up to three time for a ring, that means there's a 21.6% chance of him failing all three times, or a nearly 80% chance of success. That's good, but still says that one out of five seasons we can expect the 7th ring to not be in effect, meaning that all the other rings can potentially botch.
Now, I know that my numbers above are wild estimates. You could recite each ring for longer durations with a decent chance of success. But over time that would require more recitations to account for the inevitable failures. No doubt there's a sweet spot for rolls, probably around the 70-80% chance of success. Also, I was assuming a 9 Ars Notoria for the entire 60 years, when it was actually lower for most of that. So the lower AN score would have translated into more rolls.
I guess my point is that by my (very, very rough) estimate, Poenitens would have had to have made something around 270 rolls all with the potential to botch. Even cutting that down by 1/3 for estimation error, that's still almost 200 rolls. Given that a botch occurs about once in 100 rolls, we could expect that he would have botched about 2 times (give or take) over the course of his career under generous estimates. That's not something that's easily glossed over.
Ars Notoria has some really good benefit, to be sure. But it comes at a cost, and part of that cost is the chance for botching. And we do a disservice to game balance if we ignore the chance of botches.