Rings in CN aren't included because they max the counter, and without effort (read: purely luck) similar to Gamma's Hot Shelter. MG1, for example, allows a maximum to be displayed because it is skill based. I may disagree with the mindset itself, but it is consistent where applied, I believe.
Scores in CN are completely luck-based (you can't get the full time bonus, so you'd have to get a Perfect bonus, as well as all enemies (there are very few, and I can only think of two that are able to be combo'd together), and bumpers, which leads to a score based almost entirely on getting fast jackpots for variation). Unless I'm mistaken, un-maxable rings are the only time TSC allows luck-based records (Ring monitors in various games, I believe, are the only exception to the rule. This is outside of my realm of knowledge, and before my time, however.)
If S4 has a time over, than I agree with your assessment of CS2-3, and this should be changed immediately. I would give more opinion on CS1, but I've yet to play the game. I agree, however, with the S2 ruling for rings in CN, and am leaning yes (for that reason) for keeping scores out as well.
That said, if CN had a 4-digit counter and kept a time limit, it would be an interesting "how lucky do you feel?" chart, haha.